Solve 3ax^2+6ax-45a

$$3 a x ^ { 2 } + 6 a x - 45 a$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(ax^{2}+2ax-15a\right)$$

Consider ax^{2}+2ax-15a. Factor out a.

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

Consider x^{2}+2x-15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+px+qx-15. To find p and q, set up a system to be solved.

$$p+q=2 pq=1\left(-15\right)=-15$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$p=-3 q=5$$

Rewrite x^{2}+2x-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.

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

Evaluate

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