Solve n^3+3n^2+2n

$$n ^ { 3 } + 3 n ^ { 2 } + 2 n$$

Factor

$$n\left(n+1\right)\left(n+2\right)$$

Solution Steps

Factor out n.

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

Consider n^{2}+3n+2. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+2. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

Rewrite n^{2}+3n+2 as \left(n^{2}+n\right)+\left(2n+2\right).

$$\left(n^{2}+n\right)+\left(2n+2\right)$$

Factor out n in the first and 2 in the second group.

$$n\left(n+1\right)+2\left(n+1\right)$$

Factor out common term n+1 by using distributive property.

$$\left(n+1\right)\left(n+2\right)$$

Rewrite the complete factored expression.

$$n\left(n+1\right)\left(n+2\right)$$

Evaluate

$$n\left(n+1\right)\left(n+2\right)$$