Solve a^3+5a^2+6a

$$a ^ { 3 } + 5 a ^ { 2 } + 6 a$$

Factor

$$a\left(a+2\right)\left(a+3\right)$$

Solution Steps

Factor out a.

$$a\left(a^{2}+5a+6\right)$$

Consider a^{2}+5a+6. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+6. To find p and q, set up a system to be solved.

$$p+q=5 pq=1\times 6=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$p=2 q=3$$

Rewrite a^{2}+5a+6 as \left(a^{2}+2a\right)+\left(3a+6\right).

$$\left(a^{2}+2a\right)+\left(3a+6\right)$$

Factor out a in the first and 3 in the second group.

$$a\left(a+2\right)+3\left(a+2\right)$$

Factor out common term a+2 by using distributive property.

$$\left(a+2\right)\left(a+3\right)$$

Rewrite the complete factored expression.

$$a\left(a+2\right)\left(a+3\right)$$

Evaluate

$$a\left(a+2\right)\left(a+3\right)$$