Solve -3a(+4/3a^{3}-2/3a^2-1/6a)-6a^2(-5/3a-frac{1}{4}a^2)+frac{3}{2}(-a^4

$$- 3 a ( + \frac { 4 } { 3 } a ^ { 3 } - \frac { 2 } { 3 } a ^ { 2 } - \frac { 1 } { 6 } a ) - 6 a ^ { 2 } ( - \frac { 5 } { 3 } a - \frac { 1 } { 4 } a ^ { 2 } ) + \frac { 3 } { 2 } ( - a ^ { 4 }$$

Evaluate

$$\frac{a^{2}\left(1+24a-8a^{2}\right)}{2}$$

Solution Steps

Use the distributive property to multiply -3a by \frac{4}{3}a^{3}-\frac{2}{3}a^{2}-\frac{1}{6}a.

$$-4a^{4}+2a^{3}+\frac{1}{2}a^{2}-6a^{2}\left(-\frac{5}{3}a-\frac{1}{4}a^{2}\right)+\frac{3}{2}\left(-a^{4}\right)$$

Multiply -1 and 6 to get -6.

$$-4a^{4}+2a^{3}+\frac{1}{2}a^{2}-6a^{2}\left(-\frac{5}{3}a-\frac{1}{4}a^{2}\right)+\frac{3}{2}\left(-1\right)a^{4}$$

Use the distributive property to multiply -6a^{2} by -\frac{5}{3}a-\frac{1}{4}a^{2}.

$$-4a^{4}+2a^{3}+\frac{1}{2}a^{2}+10a^{3}+\frac{3}{2}a^{4}+\frac{3}{2}\left(-1\right)a^{4}$$

Combine 2a^{3} and 10a^{3} to get 12a^{3}.

$$-4a^{4}+12a^{3}+\frac{1}{2}a^{2}+\frac{3}{2}a^{4}+\frac{3}{2}\left(-1\right)a^{4}$$

Combine -4a^{4} and \frac{3}{2}a^{4} to get -\frac{5}{2}a^{4}.

$$-\frac{5}{2}a^{4}+12a^{3}+\frac{1}{2}a^{2}+\frac{3}{2}\left(-1\right)a^{4}$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$-\frac{5}{2}a^{4}+12a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a^{4}$$

Combine -\frac{5}{2}a^{4} and -\frac{3}{2}a^{4} to get -4a^{4}.

$$-4a^{4}+12a^{3}+\frac{1}{2}a^{2}$$

Factor

$$-4\left(a-\left(-\frac{\sqrt{38}}{4}+\frac{3}{2}\right)\right)\left(a-\left(\frac{\sqrt{38}}{4}+\frac{3}{2}\right)\right)a^{2}$$