Solve [(x+1)^{2}-4x]^{2}-(x^2+1)^2-(x^2-2x)^2

$$[ ( x + 1 ) ^ { 2 } - 4 x ] ^ { 2 } - ( x ^ { 2 } + 1 ) ^ { 2 } - ( x ^ { 2 } - 2 x ) ^ { 2 }$$

Evaluate

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

Solution Steps

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

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

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

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

Square x^{2}-2x+1.

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

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

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

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

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

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

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

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

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

Combine 6x^{2} and -2x^{2} to get 4x^{2}.

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

Subtract 1 from 1 to get 0.

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

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

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

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

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

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

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

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

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

Combine -4x^{3} and 4x^{3} to get 0.

$$4x^{2}-4x-x^{4}-4x^{2}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-4x-x^{4}$$

Expand

$$-x^{4}-4x$$

Solution Steps

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

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

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

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

Square x^{2}-2x+1.

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

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

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

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

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

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

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

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

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

Combine 6x^{2} and -2x^{2} to get 4x^{2}.

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

Subtract 1 from 1 to get 0.

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

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

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

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

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

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

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

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

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

Combine -4x^{3} and 4x^{3} to get 0.

$$4x^{2}-4x-x^{4}-4x^{2}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-4x-x^{4}$$