Solve 5x^3+45x^2+70x

$$5 x ^ { 3 } + 45 x ^ { 2 } + 70 x$$

Factor

$$5x\left(x+2\right)\left(x+7\right)$$

Solution Steps

Factor out 5.

$$5\left(x^{3}+9x^{2}+14x\right)$$

Consider x^{3}+9x^{2}+14x. Factor out x.

$$x\left(x^{2}+9x+14\right)$$

Consider x^{2}+9x+14. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 14=14$$

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 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 9.

$$a=2 b=7$$

Rewrite x^{2}+9x+14 as \left(x^{2}+2x\right)+\left(7x+14\right).

$$\left(x^{2}+2x\right)+\left(7x+14\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x+2\right)+7\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+7\right)$$

Rewrite the complete factored expression.

$$5x\left(x+2\right)\left(x+7\right)$$

Evaluate

$$5x\left(x+2\right)\left(x+7\right)$$