Solve (-x-1)^2(x-1)^2-(x+3)^2(x-3)^2

$$( - x - 1 ) ^ { 2 } ( x - 1 ) ^ { 2 } - ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 2 }$$

Evaluate

$$16\left(x^{2}-5\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-1\right)^{2}.

$$\left(\left(-x\right)^{2}-2\left(-x\right)+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Calculate -x to the power of 2 and get x^{2}.

$$\left(x^{2}-2\left(-x\right)+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Multiply -2 and -1 to get 2.

$$\left(x^{2}+2x+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}+2x+1\right)\left(x^{2}-2x+1\right)-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use the distributive property to multiply x^{2}+2x+1 by x^{2}-2x+1 and combine like terms.

$$x^{4}-2x^{2}+1-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{4}-2x^{2}+1-\left(x^{2}+6x+9\right)\left(x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{4}-2x^{2}+1-\left(x^{2}+6x+9\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply x^{2}+6x+9 by x^{2}-6x+9 and combine like terms.

$$x^{4}-2x^{2}+1-\left(x^{4}-18x^{2}+81\right)$$

To find the opposite of x^{4}-18x^{2}+81, find the opposite of each term.

$$x^{4}-2x^{2}+1-x^{4}+18x^{2}-81$$

Combine x^{4} and -x^{4} to get 0.

$$-2x^{2}+1+18x^{2}-81$$

Combine -2x^{2} and 18x^{2} to get 16x^{2}.

$$16x^{2}+1-81$$

Subtract 81 from 1 to get -80.

$$16x^{2}-80$$

Expand

$$16x^{2}-80$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-1\right)^{2}.

$$\left(\left(-x\right)^{2}-2\left(-x\right)+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Calculate -x to the power of 2 and get x^{2}.

$$\left(x^{2}-2\left(-x\right)+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Multiply -2 and -1 to get 2.

$$\left(x^{2}+2x+1\right)\left(x-1\right)^{2}-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}+2x+1\right)\left(x^{2}-2x+1\right)-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use the distributive property to multiply x^{2}+2x+1 by x^{2}-2x+1 and combine like terms.

$$x^{4}-2x^{2}+1-\left(x+3\right)^{2}\left(x-3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{4}-2x^{2}+1-\left(x^{2}+6x+9\right)\left(x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{4}-2x^{2}+1-\left(x^{2}+6x+9\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply x^{2}+6x+9 by x^{2}-6x+9 and combine like terms.

$$x^{4}-2x^{2}+1-\left(x^{4}-18x^{2}+81\right)$$

To find the opposite of x^{4}-18x^{2}+81, find the opposite of each term.

$$x^{4}-2x^{2}+1-x^{4}+18x^{2}-81$$

Combine x^{4} and -x^{4} to get 0.

$$-2x^{2}+1+18x^{2}-81$$

Combine -2x^{2} and 18x^{2} to get 16x^{2}.

$$16x^{2}+1-81$$

Subtract 81 from 1 to get -80.

$$16x^{2}-80$$