Solve x^4+8x^3+15x^2

$${ x }^{ 4 } +8 { x }^{ 3 } +15 { x }^{ 2 }$$

Factor

$$\left(x+3\right)\left(x+5\right)x^{2}$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(x^{2}+8x+15\right)$$

Consider x^{2}+8x+15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 15=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite x^{2}+8x+15 as \left(x^{2}+3x\right)+\left(5x+15\right).

$$\left(x^{2}+3x\right)+\left(5x+15\right)$$

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

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

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

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

Rewrite the complete factored expression.

$$x^{2}\left(x+3\right)\left(x+5\right)$$

Evaluate

$$\left(x+3\right)\left(x+5\right)x^{2}$$