Solve left(x^2+1right)^2-x^2+xleft(x-2right)left(x^2+x+1right)

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply x by x-2.

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

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

$$x^{4}+x^{2}+1+x^{4}-x^{3}-x^{2}-2x$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}+x^{2}+1-x^{3}-x^{2}-2x$$

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

$$2x^{4}+1-x^{3}-2x$$

Expand

$$2x^{4}-x^{3}-2x+1$$

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply x by x-2.

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

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

$$x^{4}+x^{2}+1+x^{4}-x^{3}-x^{2}-2x$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}+x^{2}+1-x^{3}-x^{2}-2x$$

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

$$2x^{4}+1-x^{3}-2x$$