Solve 5x^3+40x^2+75x

$$5 x ^ { 3 } + 40 x ^ { 2 } + 75 x$$

Factor

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

Solution Steps

Factor out 5.

$$5\left(x^{3}+8x^{2}+15x\right)$$

Consider x^{3}+8x^{2}+15x. Factor out x.

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

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.

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

Evaluate

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