Solve x^3+45x^2+86x

$$x ^ { 3 } + 45 x ^ { 2 } + 86 x$$

Factor

$$x\left(x+2\right)\left(x+43\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}+45x+86\right)$$

Consider x^{2}+45x+86. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+86. To find a and b, set up a system to be solved.

$$a+b=45 ab=1\times 86=86$$

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 86.

$$1,86 2,43$$

Calculate the sum for each pair.

$$1+86=87 2+43=45$$

The solution is the pair that gives sum 45.

$$a=2 b=43$$

Rewrite x^{2}+45x+86 as \left(x^{2}+2x\right)+\left(43x+86\right).

$$\left(x^{2}+2x\right)+\left(43x+86\right)$$

Factor out x in the first and 43 in the second group.

$$x\left(x+2\right)+43\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+43\right)$$

Rewrite the complete factored expression.

$$x\left(x+2\right)\left(x+43\right)$$

Evaluate

$$x\left(x+2\right)\left(x+43\right)$$