Solve (x-4)^{2}(x-2)^2(x)^2(x+2)^2(x+4)^2=y

$$( x - 4 ) ^ { 2 } ( x - 2 ) ^ { 2 } ( x ) ^ { 2 } ( x + 2 ) ^ { 2 } ( x + 4 ) ^ { 2 } = y$$

Solve for y

$$y=\left(x\left(x^{2}-16\right)\left(x^{2}-4\right)\right)^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$\left(x^{2}-8x+16\right)\left(x-2\right)^{2}x^{2}\left(x+2\right)^{2}\left(x+4\right)^{2}=y$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$\left(x^{2}-8x+16\right)\left(x^{2}-4x+4\right)x^{2}\left(x+2\right)^{2}\left(x+4\right)^{2}=y$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$\left(x^{2}-8x+16\right)\left(x^{2}-4x+4\right)x^{2}\left(x^{2}+4x+4\right)\left(x+4\right)^{2}=y$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$\left(x^{2}-8x+16\right)\left(x^{2}-4x+4\right)x^{2}\left(x^{2}+4x+4\right)\left(x^{2}+8x+16\right)=y$$

Use the distributive property to multiply x^{2}-8x+16 by x^{2}-4x+4 and combine like terms.

$$\left(x^{4}-12x^{3}+52x^{2}-96x+64\right)x^{2}\left(x^{2}+4x+4\right)\left(x^{2}+8x+16\right)=y$$

Use the distributive property to multiply x^{4}-12x^{3}+52x^{2}-96x+64 by x^{2}.

$$\left(x^{6}-12x^{5}+52x^{4}-96x^{3}+64x^{2}\right)\left(x^{2}+4x+4\right)\left(x^{2}+8x+16\right)=y$$

Use the distributive property to multiply x^{6}-12x^{5}+52x^{4}-96x^{3}+64x^{2} by x^{2}+4x+4 and combine like terms.

$$\left(x^{8}-8x^{7}+8x^{6}+64x^{5}-112x^{4}-128x^{3}+256x^{2}\right)\left(x^{2}+8x+16\right)=y$$

Use the distributive property to multiply x^{8}-8x^{7}+8x^{6}+64x^{5}-112x^{4}-128x^{3}+256x^{2} by x^{2}+8x+16 and combine like terms.

$$x^{10}-40x^{8}+528x^{6}-2560x^{4}+4096x^{2}=y$$

Swap sides so that all variable terms are on the left hand side.

$$y=x^{10}-40x^{8}+528x^{6}-2560x^{4}+4096x^{2}$$