Solve =(x^{3}+x^{2}+x)^{2}-(x^{2}+x^3)^2-(x^2+x)^2

$$= ( x ^ { 3 } + x ^ { 2 } + x ) ^ { 2 } - ( x ^ { 2 } + x ^ { 3 } ) ^ { 2 } - ( x ^ { 2 } + x ) ^ { 2 }$$

Evaluate

$$x^{4}$$

Solution Steps

Square x^{3}+x^{2}+x.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{2}+x^{3}\right)^{2}-\left(x^{2}+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+x^{3}\right)^{2}.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(\left(x^{2}\right)^{2}+2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

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

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{5}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{5}+x^{6}\right)-\left(x^{2}+x\right)^{2}$$

To find the opposite of x^{4}+2x^{5}+x^{6}, find the opposite of each term.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-x^{4}-2x^{5}-x^{6}-\left(x^{2}+x\right)^{2}$$

Combine 3x^{4} and -x^{4} to get 2x^{4}.

$$x^{6}+2x^{5}+2x^{4}+2x^{3}+x^{2}-2x^{5}-x^{6}-\left(x^{2}+x\right)^{2}$$

Combine 2x^{5} and -2x^{5} to get 0.

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

Combine x^{6} and -x^{6} to get 0.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$x^{4}$$

Expand

$$x^{4}$$

Solution Steps

Square x^{3}+x^{2}+x.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{2}+x^{3}\right)^{2}-\left(x^{2}+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+x^{3}\right)^{2}.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(\left(x^{2}\right)^{2}+2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

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

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{5}+\left(x^{3}\right)^{2}\right)-\left(x^{2}+x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-\left(x^{4}+2x^{5}+x^{6}\right)-\left(x^{2}+x\right)^{2}$$

To find the opposite of x^{4}+2x^{5}+x^{6}, find the opposite of each term.

$$x^{6}+2x^{5}+3x^{4}+2x^{3}+x^{2}-x^{4}-2x^{5}-x^{6}-\left(x^{2}+x\right)^{2}$$

Combine 3x^{4} and -x^{4} to get 2x^{4}.

$$x^{6}+2x^{5}+2x^{4}+2x^{3}+x^{2}-2x^{5}-x^{6}-\left(x^{2}+x\right)^{2}$$

Combine 2x^{5} and -2x^{5} to get 0.

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

Combine x^{6} and -x^{6} to get 0.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$x^{4}$$