So I want to factorize a polynomial $\Theta(x)$ which is defined below: $$\Theta(x) = (768h^8x^8-3072h^8x^7-768h^6w^2x^6+4992h^8x^6-768Hh^6x^6+2304h^6w^2x^5-4224h^8x^5+2304Hh^6x^5+288h^4w^4x^4-3264h^6w^2x^4+576Hh^4w^2x^4+1968h^8x^4-2688Hh^6x^4+192H^2h^4x^4-576h^4w^4x^3+2688h^6w^2x^3-1152Hh^4w^2x^3-480h^8x^3+1536Hh^6x^3-384H^2h^4x^3+336h^4w^4x^2-816h^6w^2x^2+672Hh^4w^2x^2-96h^8x^2+144H^2h^6x^2+48Hh^6x^2+224H^2h^4x^2-48h^4w^4x-144h^6w^2x-96Hh^4w^2x+144h^8x-144H^2h^6x-432Hh^6x-32H^2h^4x+3w^8+12Hw^6-6h^4w^4+12H^2w^4+84h^6w^2-12Hh^4w^2-36h^8+36H^2h^6+120Hh^6-4H^2h^4)$$
My guess for the factorize version of this polynomial $\Theta(x)$ is writtien below: $$3h^8(2x+\frac{w}{h}-1)(2x+\frac{w}{h}-2)(2x-\frac{w}{h})(2x-\frac{w}{h}-1)(2x-1/2+\sqrt{(\frac{w}{h}-1/2)^2+\frac{2H}{h^2}})(2x-3/2+\sqrt{(\frac{w}{h}-1/2)^2+\frac{2H}{h^2}})(2x-1/2-\sqrt{(\frac{w}{h}-1/2)^2+\frac{2H}{h^2}})(2x-3/2-\sqrt{(\frac{w}{h}-1/2)^2+\frac{2H}{h^2}})$$
So, Now If I now expand the above written product the result I get is this: $$768h^8x^8-3072h^8x^7-768h^6w^2x^6+768h^7wx^6+4992h^8x^6-768Hh^6x^6+2304h^6w^2x^5-2304h^7wx^5-4224h^8x^5+2304Hh^6x^5+288h^4w^4x^4-576h^5w^3x^4-2400h^6w^2x^4+576Hh^4w^2x^4+2688h^7wx^4-576Hh^5wx^4+1968h^8x^4-2688Hh^6x^4+192H^2h^4x^4-576h^4w^4x^3+1152h^5w^3x^3+960h^6w^2x^3-1152Hh^4w^2x^3-1536h^7wx^3+1152Hh^5wx^3-480h^8x^3+1536Hh^6x^3-384H^2h^4x^3-48h^2w^6x^2+144h^3w^5x^2+264h^4w^4x^2-144Hh^2w^4x^2-768h^5w^3x^2+288Hh^3w^3x^2-24h^6w^2x^2+672Hh^4w^2x^2-96H^2h^2w^2x^2+432h^7wx^2-816Hh^5wx^2+96H^2h^3wx^2+48h^8x^2-432Hh^6x^2+240H^2h^4x^2+48h^2w^6x-144h^3w^5x+24h^4w^4x+144Hh^2w^4x+192h^5w^3x-288Hh^3w^3x-72h^6w^2x-96Hh^4w^2x+96H^2h^2w^2x-48h^7wx+240Hh^5wx-96H^2h^3wx+48Hh^6x-48H^2h^4x+3w^8-12hw^7+6h^2w^6+12Hw^6+24h^3w^5-36Hhw^5-21h^4w^4+12H^2w^4-12h^5w^3+60Hh^3w^3-24H^2hw^3+12h^6w^2-12Hh^4w^2-12H^2h^2w^2-24Hh^5w+24H^2h^3w$$
So, you can see that the expansion of factorize version (guessed) is somewhat same to the polynomial $\Theta(x)$ that I want to factor. Now my problem is what or how should I make changes to the factorize version that I guessed so that my expansion of factorize version mathches the polynomial $\Theta(x)$.
So If I know subract the two expressions this is what I get $$(-768h^7wx^6)+2304h^7wx^5+(576h^5w^3-864h^6w^2+(576Hh^5-2688h^7)w)x^4+((-1152h^5w^3)+1728h^6w^2+(1536h^7-1152Hh^5)w)x^3+(48h^2w^6-144h^3w^5+(72h^4+144Hh^2)w^4+(768h^5-288Hh^3)w^3+(96H^2h^2-792h^6)w^2+((-432h^7)+816Hh^5-96H^2h^3)w-144h^8+(144H^2+480H)h^6-16H^2h^4)x^2+((-48h^2w^6)+144h^3w^5+((-72h^4)-144Hh^2)w^4+(288Hh^3-192h^5)w^3+((-72h^6)-96H^2h^2)w^2+(48h^7-240Hh^5+96H^2h^3)w+144h^8+((-144H^2)-480H)h^6+16H^2h^4)x+12hw^7-6h^2w^6+(36Hh-24h^3)w^5+15h^4w^4+(12h^5-60Hh^3+24H^2h)w^3+(72h^6+12H^2h^2)w^2+(24Hh^5-24H^2h^3)w-36h^8+(36H^2+120H)h^6-4H^2h^4$$
So I got this polynomial $\Theta(x)$ by doing commutator algebra. I computed some coefficients and a Casimir operator from that algebra which I know are all correct. Then I had to substitute these coeffcients and casimir operator which I caluclated using commutator algebra into a predifined function $\Theta(x)$ which is defined in terms of that Coefficients and Casimir operator. So that's how I got $\Theta(x)$. To carry out further analysis I have to write it in factorize form
I would proceed as follows.
a) simplify notation
put for instance $$ 2x = y\quad {w \over h} - {1 \over 2} = a\quad 2{H \over {h^{\,2} }} = b^{\,2} $$ and get the polynomial in $y$ and two parameters $$ p(y,a,b) = 3h^{\,8} \sum\limits_{k = 0}^8 {f_{\,k} \left( {a,b} \right)y^{\,k} } = 3h^{\,8} \prod\limits_{j = 1}^8 {\left( {y - r_{\,j} (a,b)} \right)} $$
where the roots are your attempted roots.
b) pass to leading coeff. $1$
$$ q(y,a,b) = {{p(y,a,b)} \over {3h^{\,8} }} = \sum\limits_{k = 0}^8 {f_{\,k} \left( {a,b} \right)y^{\,k} } = \prod\limits_{j = 1}^8 {\left( {y - r_{\,j} (a,b)} \right)} $$
c) check for $y=r_j(a,b)$
check each supposed root to give $0$ when replaced for $y$ in the sum version;
take note of the valid and invalid roots;
d) divide by the valid roots
perform long division to get a a polynomial of reduced degree, containing only the invalid roots.
e) check for $y=0$ in the reduced "invalid" polynomial
you should get $$ f_{\,0} \left( {a,b} \right) = \left( { - 1} \right)^{\,n} \prod\limits_{j = 1}^n {\left( {y - r_{\,j} (a,b)} \right)} $$ try and see what goes wrong, also by putting some easy values (0,1, ..) for $a,b$;
try and get some other valid root and repeat from d).