Solve (x-1)(x^{2}+x)^2-(x^3-5x^4+x^3)

$$( x - 1 ) ( x ^ { 2 } + x ) ^ { 2 } - ( x ^ { 3 } - 5 x ^ { 4 } + x ^ { 3 } )$$

Evaluate

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

Solution Steps

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

$$\left(x-1\right)\left(\left(x^{2}\right)^{2}+2x^{2}x+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

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

$$\left(x-1\right)\left(x^{4}+2x^{2}x+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

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

$$\left(x-1\right)\left(x^{4}+2x^{3}+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

Use the distributive property to multiply x-1 by x^{4}+2x^{3}+x^{2} and combine like terms.

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

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

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

To find the opposite of 2x^{3}-5x^{4}, find the opposite of each term.

$$x^{5}+x^{4}-x^{3}-x^{2}-2x^{3}+5x^{4}$$

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

$$x^{5}+x^{4}-3x^{3}-x^{2}+5x^{4}$$

Combine x^{4} and 5x^{4} to get 6x^{4}.

$$x^{5}+6x^{4}-3x^{3}-x^{2}$$

Expand

$$x^{5}+6x^{4}-3x^{3}-x^{2}$$

Solution Steps

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

$$\left(x-1\right)\left(\left(x^{2}\right)^{2}+2x^{2}x+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

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

$$\left(x-1\right)\left(x^{4}+2x^{2}x+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

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

$$\left(x-1\right)\left(x^{4}+2x^{3}+x^{2}\right)-\left(x^{3}-5x^{4}+x^{3}\right)$$

Use the distributive property to multiply x-1 by x^{4}+2x^{3}+x^{2} and combine like terms.

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

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

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

To find the opposite of 2x^{3}-5x^{4}, find the opposite of each term.

$$x^{5}+x^{4}-x^{3}-x^{2}-2x^{3}+5x^{4}$$

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

$$x^{5}+x^{4}-3x^{3}-x^{2}+5x^{4}$$

Combine x^{4} and 5x^{4} to get 6x^{4}.

$$x^{5}+6x^{4}-3x^{3}-x^{2}$$