Solve (a-4)(a^2/4+a+4)+3(a^2-a-1)(a-3)+7

$$( a - 4 ) ( \frac { a ^ { 2 } } { 4 } + a + 4 ) + 3 ( a ^ { 2 } - a - 1 ) ( a - 3 ) + 7$$

Evaluate

$$\frac{13a^{3}}{4}-12a^{2}+6a$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a+4 times \frac{4}{4}.

$$\left(a-4\right)\left(\frac{a^{2}}{4}+\frac{4\left(a+4\right)}{4}\right)+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Since \frac{a^{2}}{4} and \frac{4\left(a+4\right)}{4} have the same denominator, add them by adding their numerators.

$$\left(a-4\right)\times \frac{a^{2}+4\left(a+4\right)}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Do the multiplications in a^{2}+4\left(a+4\right).

$$\left(a-4\right)\times \frac{a^{2}+4a+16}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Express \left(a-4\right)\times \frac{a^{2}+4a+16}{4} as a single fraction.

$$\frac{\left(a-4\right)\left(a^{2}+4a+16\right)}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Use the distributive property to multiply a-4 by a^{2}+4a+16 and combine like terms.

$$\frac{a^{3}-64}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

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

$$\frac{a^{3}-64}{4}+\left(3a^{2}-3a-3\right)\left(a-3\right)+7$$

Use the distributive property to multiply 3a^{2}-3a-3 by a-3 and combine like terms.

$$\frac{a^{3}-64}{4}+3a^{3}-12a^{2}+6a+9+7$$

Add 9 and 7 to get 16.

$$\frac{a^{3}-64}{4}+3a^{3}-12a^{2}+6a+16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3a^{3}-12a^{2}+6a+16 times \frac{4}{4}.

$$\frac{a^{3}-64}{4}+\frac{4\left(3a^{3}-12a^{2}+6a+16\right)}{4}$$

Since \frac{a^{3}-64}{4} and \frac{4\left(3a^{3}-12a^{2}+6a+16\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{a^{3}-64+4\left(3a^{3}-12a^{2}+6a+16\right)}{4}$$

Do the multiplications in a^{3}-64+4\left(3a^{3}-12a^{2}+6a+16\right).

$$\frac{a^{3}-64+12a^{3}-48a^{2}+24a+64}{4}$$

Combine like terms in a^{3}-64+12a^{3}-48a^{2}+24a+64.

$$\frac{13a^{3}-48a^{2}+24a}{4}$$

Expand

$$\frac{13a^{3}}{4}-12a^{2}+6a$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a+4 times \frac{4}{4}.

$$\left(a-4\right)\left(\frac{a^{2}}{4}+\frac{4\left(a+4\right)}{4}\right)+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Since \frac{a^{2}}{4} and \frac{4\left(a+4\right)}{4} have the same denominator, add them by adding their numerators.

$$\left(a-4\right)\times \frac{a^{2}+4\left(a+4\right)}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Do the multiplications in a^{2}+4\left(a+4\right).

$$\left(a-4\right)\times \frac{a^{2}+4a+16}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Express \left(a-4\right)\times \frac{a^{2}+4a+16}{4} as a single fraction.

$$\frac{\left(a-4\right)\left(a^{2}+4a+16\right)}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

Use the distributive property to multiply a-4 by a^{2}+4a+16 and combine like terms.

$$\frac{a^{3}-64}{4}+3\left(a^{2}-a-1\right)\left(a-3\right)+7$$

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

$$\frac{a^{3}-64}{4}+\left(3a^{2}-3a-3\right)\left(a-3\right)+7$$

Use the distributive property to multiply 3a^{2}-3a-3 by a-3 and combine like terms.

$$\frac{a^{3}-64}{4}+3a^{3}-12a^{2}+6a+9+7$$

Add 9 and 7 to get 16.

$$\frac{a^{3}-64}{4}+3a^{3}-12a^{2}+6a+16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3a^{3}-12a^{2}+6a+16 times \frac{4}{4}.

$$\frac{a^{3}-64}{4}+\frac{4\left(3a^{3}-12a^{2}+6a+16\right)}{4}$$

Since \frac{a^{3}-64}{4} and \frac{4\left(3a^{3}-12a^{2}+6a+16\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{a^{3}-64+4\left(3a^{3}-12a^{2}+6a+16\right)}{4}$$

Do the multiplications in a^{3}-64+4\left(3a^{3}-12a^{2}+6a+16\right).

$$\frac{a^{3}-64+12a^{3}-48a^{2}+24a+64}{4}$$

Combine like terms in a^{3}-64+12a^{3}-48a^{2}+24a+64.

$$\frac{13a^{3}-48a^{2}+24a}{4}$$