Solve (8)(x^2+x)^2-(x+1)^2

$$( 8 ) ( x ^ { 2 } + x ) ^ { 2 } - ( x + 1 ) ^ { 2 }$$

Evaluate

$$\left(8x^{2}-1\right)\left(x+1\right)^{2}$$

Solution Steps

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

$$8\left(\left(x^{2}\right)^{2}+2x^{2}x+x^{2}\right)-\left(x+1\right)^{2}$$

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

$$8\left(x^{4}+2x^{2}x+x^{2}\right)-\left(x+1\right)^{2}$$

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

$$8\left(x^{4}+2x^{3}+x^{2}\right)-\left(x+1\right)^{2}$$

Use the distributive property to multiply 8 by x^{4}+2x^{3}+x^{2}.

$$8x^{4}+16x^{3}+8x^{2}-\left(x+1\right)^{2}$$

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

$$8x^{4}+16x^{3}+8x^{2}-\left(x^{2}+2x+1\right)$$

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

$$8x^{4}+16x^{3}+8x^{2}-x^{2}-2x-1$$

Combine 8x^{2} and -x^{2} to get 7x^{2}.

$$8x^{4}+16x^{3}+7x^{2}-2x-1$$

Expand

$$8x^{4}+16x^{3}+7x^{2}-2x-1$$

Solution Steps

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

$$8\left(\left(x^{2}\right)^{2}+2x^{2}x+x^{2}\right)-\left(x+1\right)^{2}$$

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

$$8\left(x^{4}+2x^{2}x+x^{2}\right)-\left(x+1\right)^{2}$$

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

$$8\left(x^{4}+2x^{3}+x^{2}\right)-\left(x+1\right)^{2}$$

Use the distributive property to multiply 8 by x^{4}+2x^{3}+x^{2}.

$$8x^{4}+16x^{3}+8x^{2}-\left(x+1\right)^{2}$$

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

$$8x^{4}+16x^{3}+8x^{2}-\left(x^{2}+2x+1\right)$$

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

$$8x^{4}+16x^{3}+8x^{2}-x^{2}-2x-1$$

Combine 8x^{2} and -x^{2} to get 7x^{2}.

$$8x^{4}+16x^{3}+7x^{2}-2x-1$$