Solve 2x^3+13x^2+15x

$$2 { x }^{ 3 } +13 { x }^{ 2 } +15x$$

Factor

$$x\left(x+5\right)\left(2x+3\right)$$

Solution Steps

Factor out x.

$$x\left(2x^{2}+13x+15\right)$$

Consider 2x^{2}+13x+15. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=13 ab=2\times 15=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=3 b=10$$

Rewrite 2x^{2}+13x+15 as \left(2x^{2}+3x\right)+\left(10x+15\right).

$$\left(2x^{2}+3x\right)+\left(10x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(2x+3\right)+5\left(2x+3\right)$$

Factor out common term 2x+3 by using distributive property.

$$\left(2x+3\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$x\left(2x+3\right)\left(x+5\right)$$

Evaluate

$$x\left(x+5\right)\left(2x+3\right)$$