Solve 3m^3+3m^2-126m

$$3 m ^ { 3 } + 3 m ^ { 2 } - 126 m$$

Factor

$$3m\left(m-6\right)\left(m+7\right)$$

Solution Steps

Factor out 3.

$$3\left(m^{3}+m^{2}-42m\right)$$

Consider m^{3}+m^{2}-42m. Factor out m.

$$m\left(m^{2}+m-42\right)$$

Consider m^{2}+m-42. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm-42. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-42\right)=-42$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -42.

$$-1,42 -2,21 -3,14 -6,7$$

Calculate the sum for each pair.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

The solution is the pair that gives sum 1.

$$a=-6 b=7$$

Rewrite m^{2}+m-42 as \left(m^{2}-6m\right)+\left(7m-42\right).

$$\left(m^{2}-6m\right)+\left(7m-42\right)$$

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

$$m\left(m-6\right)+7\left(m-6\right)$$

Factor out common term m-6 by using distributive property.

$$\left(m-6\right)\left(m+7\right)$$

Rewrite the complete factored expression.

$$3m\left(m-6\right)\left(m+7\right)$$

Evaluate

$$3m\left(m-6\right)\left(m+7\right)$$