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

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

Evaluate

$$-2x^{2}$$

Solution Steps

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

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

Consider \left(x^{2}+x-1\right)\left(x^{2}+x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}, where a=x^{2}+x and b=1. Square 1.

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

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

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

To find the opposite of x^{4}+2x^{3}+x^{2}-1, find the opposite of each term.

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

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

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

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

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

Add -1 and 1 to get 0.

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

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

$$-2x^{2}-2x-2x^{3}+2x^{3}+2x$$

Combine -2x^{3} and 2x^{3} to get 0.

$$-2x^{2}-2x+2x$$

Combine -2x and 2x to get 0.

$$-2x^{2}$$

Expand

$$-2x^{2}$$

Solution Steps

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

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

Consider \left(x^{2}+x-1\right)\left(x^{2}+x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}, where a=x^{2}+x and b=1. Square 1.

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

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

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

To find the opposite of x^{4}+2x^{3}+x^{2}-1, find the opposite of each term.

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

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

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

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

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

Add -1 and 1 to get 0.

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

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

$$-2x^{2}-2x-2x^{3}+2x^{3}+2x$$

Combine -2x^{3} and 2x^{3} to get 0.

$$-2x^{2}-2x+2x$$

Combine -2x and 2x to get 0.

$$-2x^{2}$$