Solve (x^2-x+1)(2+x)-x(x^2-1)+1/2(x-2)(x+2)

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

Evaluate

$$\frac{3x^{2}}{2}$$

Solution Steps

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

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

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

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

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

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

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

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

Combine -x and x to get 0.

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

Use the distributive property to multiply \frac{1}{2} by x-2.

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

Use the distributive property to multiply \frac{1}{2}x-1 by x+2 and combine like terms.

$$x^{2}+2+\frac{1}{2}x^{2}-2$$

Combine x^{2} and \frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}+2-2$$

Subtract 2 from 2 to get 0.

$$\frac{3}{2}x^{2}$$

Expand

$$\frac{3x^{2}}{2}$$

Solution Steps

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

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

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

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

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

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

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

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

Combine -x and x to get 0.

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

Use the distributive property to multiply \frac{1}{2} by x-2.

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

Use the distributive property to multiply \frac{1}{2}x-1 by x+2 and combine like terms.

$$x^{2}+2+\frac{1}{2}x^{2}-2$$

Combine x^{2} and \frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}+2-2$$

Subtract 2 from 2 to get 0.

$$\frac{3}{2}x^{2}$$