Solve (a+a^2-1)*(x-1)-[a^2+x^2+ax+a^2*(x-2)]+x*(x+1)

$$( a + a ^ { 2 } - 1 ) \cdot ( x - 1 ) - [ a ^ { 2 } + x ^ { 2 } + a x + a ^ { 2 } \cdot ( x - 2 ) ] + x \cdot ( x + 1 )$$

Evaluate

$$1-a$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine ax and -ax to get 0.

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

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

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

Use the distributive property to multiply x by x+1.

$$-a-x+1-x^{2}+x^{2}+x$$

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

$$-a-x+1+x$$

Combine -x and x to get 0.

$$-a+1$$

Expand

$$1-a$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine ax and -ax to get 0.

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

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

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

Use the distributive property to multiply x by x+1.

$$-a-x+1-x^{2}+x^{2}+x$$

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

$$-a-x+1+x$$

Combine -x and x to get 0.

$$-a+1$$