Solve abx^2-5abx-24ab

$$a b x ^ { 2 } - 5 a b x - 24 a b$$

Factor

$$ab\left(x-8\right)\left(x+3\right)$$

Solution Steps

Factor out ab.

$$ab\left(x^{2}-5x-24\right)$$

Consider x^{2}-5x-24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+px+qx-24. To find p and q, set up a system to be solved.

$$p+q=-5 pq=1\left(-24\right)=-24$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$p=-8 q=3$$

Rewrite x^{2}-5x-24 as \left(x^{2}-8x\right)+\left(3x-24\right).

$$\left(x^{2}-8x\right)+\left(3x-24\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-8\right)+3\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$ab\left(x-8\right)\left(x+3\right)$$

Evaluate

$$ab\left(x-8\right)\left(x+3\right)$$