Solve a^2+4a+8a+32

$$a ^ { 2 } + 4 a + 8 a + 32$$

Factor

$$\left(a+4\right)\left(a+8\right)$$

Solution Steps

Multiply and combine like terms.

$$a^{2}+12a+32$$

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+32. To find p and q, set up a system to be solved.

$$p+q=12 pq=1\times 32=32$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 12.

$$p=4 q=8$$

Rewrite a^{2}+12a+32 as \left(a^{2}+4a\right)+\left(8a+32\right).

$$\left(a^{2}+4a\right)+\left(8a+32\right)$$

Factor out a in the first and 8 in the second group.

$$a\left(a+4\right)+8\left(a+4\right)$$

Factor out common term a+4 by using distributive property.

$$\left(a+4\right)\left(a+8\right)$$

Evaluate

$$\left(a+4\right)\left(a+8\right)$$

Solution Steps

Combine 4a and 8a to get 12a.

$$a^{2}+12a+32$$