Rrerku

I suggest something line the following, so the wide terms are reduced.

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools

begindocument

begingather*
beginalign*
g(eta,z)&=
parbox[t]0.85displaywidthraggedright
$-306772802511648469920eta^4z^4+
762453974480763801600eta^5z^4-
1678626210368271790080eta^5z^3-
28510918043555533736160eta^4z^3+
11443138641451067779872eta^3z^3-
52164076923190540413504eta^2z^2-
78145258181161076156160eta^5z^2-
211306163712129371808450eta^4z^2+
228927087397104405937944eta^3z^2+
999881065017543109136462eta^3z-
317254092617698017425280eta^5z-
443761561344388063474665eta^4z+
82327155732241730770824eta z-
514623285385260545505123eta^2z-
1010535343560043404912120eta^2-
357788302700438191196160eta^5-
43808044579418934376632-
214023244873618345872240eta^4+
11818373349781028079eta^3+
347370177721463765064153eta$

\[2ex]
h(z)&=(417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)
endalign*
\[2ex]
f(z)=frac1382112640fracg(eta,z)h(z)
endgather*

enddocument

I would use something like this

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools
begindocument
Blah blah
[f(z)=frac1382112640cdotfracAB]
where
beginalign*
 A=&,-306772802511648469920eta^4z^4+762453974480763801600eta^5z^4\
 &,-1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3\
 &,+11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2\
 &,-78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2\
 &,+228927087397104405937944eta^3z^2+999881065017543109136462eta^3z\
 &,-317254092617698017425280eta^5z-443761561344388063474665eta^4z\
 &,+82327155732241730770824eta z-514623285385260545505123eta^2z\
 &,-1010535343560043404912120eta^2-357788302700438191196160eta^5\
 &,-43808044579418934376632-214023244873618345872240eta^4\
 &,+11818373349781028079eta^3+347370177721463765064153eta
endalign*
and
[B=(417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)]
enddocument

or

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools, nccmath
begindocument
 beginmultline*medmath
f(z)=frac1382112640
 fracleft[
 beginmultlined
 -306772802511648469920eta^4z^4+762453974480763801600eta^5z^4-\
 1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3+\
 11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2-\
 78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2+\
 228927087397104405937944eta^3z^2+999881065017543109136462eta^3z-\
 317254092617698017425280eta^5z-443761561344388063474665eta^4z+\
 82327155732241730770824eta z - 514623285385260545505123eta^2z-\
 1010535343560043404912120eta^2-357788302700438191196160eta^5-\
 43808044579418934376632-214023244873618345872240eta^4+\
 11818373349781028079eta^3+347370177721463765064153eta
 endmultlinedright]
 (417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)
 endmultline*

Given the nature of the operations, you can probably express this in a tidy manner using matrix multiplication notation, eg:

where:

Code:

$$ f(z)=frac1382112640 fracg(eta, z)u(z) , v(z) , w(z) $$

where

$$
beginarrayll
g(eta, z) = beginbmatrix 
beginarrayr @hspace0em l
- & 306772802511648469920 \
 & 762453974480763801600 \
- & 1678626210368271790080 \
- & 28510918043555533736160 \
 & 11443138641451067779872 \
- & 52164076923190540413504 \
- & 78145258181161076156160 \
- & 211306163712129371808450 \
 & 228927087397104405937944 \
 & 999881065017543109136462 \
- & 317254092617698017425280 \
- & 443761561344388063474665 \
 & 82327155732241730770824 \
- & 514623285385260545505123 \
- & 1010535343560043404912120 \
- & 357788302700438191196160 \
- & 43808044579418934376632 \
- & 214023244873618345872240 \
 & 11818373349781028079 \
 & 347370177721463765064153
endarray
endbmatrix^T
beginbmatrix
eta^4z^4 \
eta^5z^4 \
eta^5z^3 \
eta^4z^3 \
eta^3z^3 \
eta^2z^2 \
eta^5z^2 \
eta^4z^2 \
eta^3z^2 \
eta^3z \
eta^5z \
eta^4z \
eta z \
eta^2z \
eta^2 \
eta^5 \
1 \
eta^4 \
eta^3 \
eta
endbmatrix & 
beginarrayl
u(z) = left ( beginbmatrix 417420 \ -4169121 \ -15571312 endbmatrix^T beginbmatrix z^2 \ z \ 1 endbmatrix right ) \[3em]
v(z) = left ( beginbmatrix 1546 \ 3537 endbmatrix^T beginbmatrix z \ 1 endbmatrix right ) \[3em]
w(z) = left ( beginbmatrix 3092 \ 17001endbmatrix^T beginbmatrix z \ 1 endbmatrix right ) \[3em]
endarray
endarray
$$

I recommend aligning the variables and adding some form of thousand-separators, both will enhance the readability. What I also recommend (but didn't do here) is sorting by the powers of the first and then the second variable. This is a modification of JuleV's answer.

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools
begindocument
Blah blah
[f(z)=frac1382112640cdotfracAB]
where
[
arraycolsep=0.5pt
beginarrayrrllrll
 A=&, -306,772,802,511,648,469,920 &eta^4 &z^4 & +762,453,974,480,763,801,600 &eta^5 &z^4\
 &, -1,678,626,210,368,271,790,080 &eta^5 &z^3 & -2,8510,918,043,555,533,736,160 &eta^4 &z^3\
 &, +11,443,138,641,451,067,779,872 &eta^3 &z^3 & -5,2164,076,923,190,540,413,504 &eta^2 &z^2\
 &, -78,145,258,181,161,076,156,160 &eta^5 &z^2 & -21,1306,163,712,129,371,808,450 &eta^4 &z^2\
 &, +228,927,087,397,104,405,937,944 &eta^3 &z^2 & +99,9881,065,017,543,109,136,462 &eta^3 &z\
 &, -317,254,092,617,698,017,425,280 &eta^5 &z & -44,3761,561,344,388,063,474,665 &eta^4 &z\
 &, +82,327,155,732,241,730,770,824 &eta &z & -51,4623,285,385,260,545,505,123 &eta^2 &z\
 &,-1,010,535,343,560,043,404,912,120 &eta^2 & & -35,7788,302,700,438,191,196,160 &eta^5 &\
 &, -43,808,044,579,418,934,376,632 & & & -21,4023,244,873,618,345,872,240 &eta^4 &\
 &, +11,818,373,349,781,028,079 &eta^3 & & +34,7370,177,721,463,765,064,153 &eta &
endarray
]
and
[B=(417,420z^2-4,169,121z-15,571,312)(1,546z+3,537)(3,092z+17,001)]
enddocument

I'm sure there are also some custom packages that can do this for you but this is just using the packages you provided:

With the original disposition of the polynomial, but using alignat, parenthesis, and fractions to emphasize the different terms of the polynomial.

documentclassarticle

usepackagemathtools

begindocument

beginalignat*2
& f(z) && = frac1382112640 times left( vphantomfrac1382112640 -306772802511648469920 eta^4 z^4 + 762453974480763801600 eta^5 z^4 right. \[1.5ex] 
& && -1678626210368271790080 eta^5 z^3 - 28510918043555533736160 eta^4 z^3 \[1.5ex]
& && +11443138641451067779872 eta^3 z^3 -52164076923190540413504 eta^2 z^2 \[1.5ex]
& && -78145258181161076156160 eta^5 z^2 -211306163712129371808450 eta^4 z^2 \[1.5ex]
& && +228927087397104405937944 eta^3 z^2 +999881065017543109136462 eta^3 z \[1.5ex]
& && -317254092617698017425280 eta^5 z -443761561344388063474665 eta^4 z \[1.5ex]
& && +82327155732241730770824 eta z -514623285385260545505123 eta^2 z \[1.5ex]
& && -1010535343560043404912120 eta^2 -357788302700438191196160 eta^5 \[1.5ex]
& && -43808044579418934376632 - 214023244873618345872240 eta^4 \[1.5ex]
& && +11818373349781028079 eta^3 +347370177721463765064153eta left. vphantomfrac1382112640 right) times frac417420z^2-4169121z-15571312(1546z+3537)(3092z+17001)
endalignat*

I would usually use the package breqn. That automatically line-breaks equations, and has a lot of very nice features, but uses low-level having into the maths primitives, which means it tends to make a mess of other packages what do the same thing (for example, you can't use both breqn and sansmath in the same document)

begindmath*
f(z)=frac1382112640times-306772802511648469920eta^4z^4+left(762453974480763801600eta^5z^4-1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3+11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2-78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2+228927087397104405937944eta^3z^2+999881065017543109136462eta^3z-317254092617698017425280eta^5z-443761561344388063474665eta^4z+82327155732241730770824eta z-514623285385260545505123eta^2z-1010535343560043404912120eta^2-357788302700438191196160eta^5-43808044579418934376632-214023244873618345872240eta^4+11818373349781028079eta^3+347370177721463765064153etaright)timesleft(left(417420z^2-4169121z-15571312right)left(1546z+3537right)left(3092z+17001right)right)^-1
enddmath*

which produces

IMO the right-alignment is ugly but apparently that's the AMS standard - without the brackets it left aligns all those lines like the alginat version.