Solve (4a^3-1/2a^2)(1/2a+3)+2a(-a^3+a)

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

Evaluate

$$\frac{\left(47a+2\right)a^{2}}{4}$$

Solution Steps

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

$$2a^{4}+\frac{47}{4}a^{3}-\frac{3}{2}a^{2}+2a\left(-a^{3}+a\right)$$

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

$$2a^{4}+\frac{47}{4}a^{3}-\frac{3}{2}a^{2}+2a\left(-a^{3}\right)+2a^{2}$$

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

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}+2a\left(-a^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}+2a^{4}\left(-1\right)$$

Multiply 2 and -1 to get -2.

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}-2a^{4}$$

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

$$\frac{47}{4}a^{3}+\frac{1}{2}a^{2}$$

Expand

$$\frac{47a^{3}}{4}+\frac{a^{2}}{2}$$

Solution Steps

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

$$2a^{4}+\frac{47}{4}a^{3}-\frac{3}{2}a^{2}+2a\left(-a^{3}+a\right)$$

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

$$2a^{4}+\frac{47}{4}a^{3}-\frac{3}{2}a^{2}+2a\left(-a^{3}\right)+2a^{2}$$

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

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}+2a\left(-a^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}+2a^{4}\left(-1\right)$$

Multiply 2 and -1 to get -2.

$$2a^{4}+\frac{47}{4}a^{3}+\frac{1}{2}a^{2}-2a^{4}$$

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

$$\frac{47}{4}a^{3}+\frac{1}{2}a^{2}$$