Solve [1/2x^2+1/4(1/2x-2)(x^2+4x-16)-(1/2x+2)+(x+3)^2]*[4x+9]

$$[ \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 4 } ( \frac { 1 } { 2 } x - 2 ) ( x ^ { 2 } + 4 x - 16 ) - ( \frac { 1 } { 2 } x + 2 ) + ( x + 3 ) ^ { 2 } ] \cdot [ 4 x + 9 ]$$

Evaluate

$$\frac{\left(4x+9\right)\left(x^{3}+12x^{2}+12x+120\right)}{8}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by \frac{1}{2}x-2.

$$\left(\frac{1}{2}x^{2}+\left(\frac{1}{8}x-\frac{1}{2}\right)\left(x^{2}+4x-16\right)-\left(\frac{1}{2}x+2\right)+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Use the distributive property to multiply \frac{1}{8}x-\frac{1}{2} by x^{2}+4x-16 and combine like terms.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-4x+8-\left(\frac{1}{2}x+2\right)+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

To find the opposite of \frac{1}{2}x+2, find the opposite of each term.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-4x+8-\frac{1}{2}x-2+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Combine -4x and -\frac{1}{2}x to get -\frac{9}{2}x.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+8-2+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Subtract 2 from 8 to get 6.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+x^{2}+6x+9\right)\left(4x+9\right)$$

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

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+6x+9\right)\left(4x+9\right)$$

Combine -\frac{9}{2}x and 6x to get \frac{3}{2}x.

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+6+9\right)\left(4x+9\right)$$

Add 6 and 9 to get 15.

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+15\right)\left(4x+9\right)$$

Use the distributive property to multiply \frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+15 by 4x+9 and combine like terms.

$$\frac{57}{8}x^{3}+\frac{39}{2}x^{2}+\frac{1}{2}x^{4}+\frac{147}{2}x+135$$

Expand

$$\frac{x^{4}}{2}+\frac{57x^{3}}{8}+\frac{39x^{2}}{2}+\frac{147x}{2}+135$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by \frac{1}{2}x-2.

$$\left(\frac{1}{2}x^{2}+\left(\frac{1}{8}x-\frac{1}{2}\right)\left(x^{2}+4x-16\right)-\left(\frac{1}{2}x+2\right)+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Use the distributive property to multiply \frac{1}{8}x-\frac{1}{2} by x^{2}+4x-16 and combine like terms.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-4x+8-\left(\frac{1}{2}x+2\right)+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

To find the opposite of \frac{1}{2}x+2, find the opposite of each term.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-4x+8-\frac{1}{2}x-2+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Combine -4x and -\frac{1}{2}x to get -\frac{9}{2}x.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+8-2+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Subtract 2 from 8 to get 6.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+\left(x+3\right)^{2}\right)\left(4x+9\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$\left(\frac{1}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+x^{2}+6x+9\right)\left(4x+9\right)$$

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

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}-\frac{9}{2}x+6+6x+9\right)\left(4x+9\right)$$

Combine -\frac{9}{2}x and 6x to get \frac{3}{2}x.

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+6+9\right)\left(4x+9\right)$$

Add 6 and 9 to get 15.

$$\left(\frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+15\right)\left(4x+9\right)$$

Use the distributive property to multiply \frac{3}{2}x^{2}+\frac{1}{8}x^{3}+\frac{3}{2}x+15 by 4x+9 and combine like terms.

$$\frac{57}{8}x^{3}+\frac{39}{2}x^{2}+\frac{1}{2}x^{4}+\frac{147}{2}x+135$$