Solve 4+1/2(x-2)(-1/2x^2+3x+4)+1/2(6-x)(-1/2x^2+3x)

$$4+ \frac{ 1 }{ 2 } (x-2)(- \frac{ 1 }{ 2 } { x }^{ 2 } +3x+4)+ \frac{ 1 }{ 2 } (6-x)(- \frac{ 1 }{ 2 } { x }^{ 2 } +3x)$$

Evaluate

$$-x\left(x-8\right)$$

Solution Steps

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

$$4+\left(\frac{1}{2}x-1\right)\left(-\frac{1}{2}x^{2}+3x+4\right)+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

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

$$4-\frac{1}{4}x^{3}+2x^{2}-x-4+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

Subtract 4 from 4 to get 0.

$$-\frac{1}{4}x^{3}+2x^{2}-x+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

Use the distributive property to multiply \frac{1}{2} by 6-x.

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

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

$$-\frac{1}{4}x^{3}+2x^{2}-x-3x^{2}+9x+\frac{1}{4}x^{3}$$

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

$$-\frac{1}{4}x^{3}-x^{2}-x+9x+\frac{1}{4}x^{3}$$

Combine -x and 9x to get 8x.

$$-\frac{1}{4}x^{3}-x^{2}+8x+\frac{1}{4}x^{3}$$

Combine -\frac{1}{4}x^{3} and \frac{1}{4}x^{3} to get 0.

$$-x^{2}+8x$$

Expand

$$8x-x^{2}$$

Solution Steps

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

$$4+\left(\frac{1}{2}x-1\right)\left(-\frac{1}{2}x^{2}+3x+4\right)+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

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

$$4-\frac{1}{4}x^{3}+2x^{2}-x-4+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

Subtract 4 from 4 to get 0.

$$-\frac{1}{4}x^{3}+2x^{2}-x+\frac{1}{2}\left(6-x\right)\left(-\frac{1}{2}x^{2}+3x\right)$$

Use the distributive property to multiply \frac{1}{2} by 6-x.

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

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

$$-\frac{1}{4}x^{3}+2x^{2}-x-3x^{2}+9x+\frac{1}{4}x^{3}$$

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

$$-\frac{1}{4}x^{3}-x^{2}-x+9x+\frac{1}{4}x^{3}$$

Combine -x and 9x to get 8x.

$$-\frac{1}{4}x^{3}-x^{2}+8x+\frac{1}{4}x^{3}$$

Combine -\frac{1}{4}x^{3} and \frac{1}{4}x^{3} to get 0.

$$-x^{2}+8x$$