Solve left(x+1right)^2-left(x+2right)^2+left(x-1right)left(x+1right)left(x^2-1right)+left(x^2-1right)left(-{x}^2+1right)

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

Evaluate

$$-2x-3$$

Solution Steps

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

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

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

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

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

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

Combine x^{2} and -x^{2} to get 0.

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

Combine 2x and -4x to get -2x.

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

Subtract 4 from 1 to get -3.

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

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

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

Multiply x^{2}-1 and x^{2}-1 to get \left(x^{2}-1\right)^{2}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Add -3 and 1 to get -2.

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

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

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

Multiply -1 and -1 to get 1.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine -2x^{2} and 2x^{2} to get 0.

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

Subtract 1 from -2 to get -3.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Combine x^{4} and x^{4}\left(-1\right) to get 0.

$$-2x-3$$

Expand

$$-2x-3$$

Solution Steps

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

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

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

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

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

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

Combine x^{2} and -x^{2} to get 0.

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

Combine 2x and -4x to get -2x.

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

Subtract 4 from 1 to get -3.

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

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

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

Multiply x^{2}-1 and x^{2}-1 to get \left(x^{2}-1\right)^{2}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Add -3 and 1 to get -2.

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

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

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

Multiply -1 and -1 to get 1.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine -2x^{2} and 2x^{2} to get 0.

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

Subtract 1 from -2 to get -3.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Combine x^{4} and x^{4}\left(-1\right) to get 0.

$$-2x-3$$