Solve (x-a)(x^{2}+ax+a^2)-a^2x-(x+a)(x-a)(x-1)+a^2(a-3)+(2a-x)^2

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

Evaluate

$$2x\left(x-2a\right)$$

Solution Steps

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

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

Use the distributive property to multiply x+a by x-a and combine like terms.

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

Use the distributive property to multiply x^{2}-a^{2} by x-1.

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

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

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

Use the distributive property to multiply a^{2} by a-3.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine -a^{3} and a^{3} to get 0.

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

Expand

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

Solution Steps

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

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

Use the distributive property to multiply x+a by x-a and combine like terms.

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

Use the distributive property to multiply x^{2}-a^{2} by x-1.

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

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

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

Use the distributive property to multiply a^{2} by a-3.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine -a^{3} and a^{3} to get 0.

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