Solve 4m^3+10m^2-84m

$$4 m ^ { 3 } + 10 m ^ { 2 } - 84 m$$

Factor

$$2m\left(2m-7\right)\left(m+6\right)$$

Solution Steps

Factor out 2.

$$2\left(2m^{3}+5m^{2}-42m\right)$$

Consider 2m^{3}+5m^{2}-42m. Factor out m.

$$m\left(2m^{2}+5m-42\right)$$

Consider 2m^{2}+5m-42. Factor the expression by grouping. First, the expression needs to be rewritten as 2m^{2}+am+bm-42. To find a and b, set up a system to be solved.

$$a+b=5 ab=2\left(-42\right)=-84$$

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 -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 5.

$$a=-7 b=12$$

Rewrite 2m^{2}+5m-42 as \left(2m^{2}-7m\right)+\left(12m-42\right).

$$\left(2m^{2}-7m\right)+\left(12m-42\right)$$

Factor out m in the first and 6 in the second group.

$$m\left(2m-7\right)+6\left(2m-7\right)$$

Factor out common term 2m-7 by using distributive property.

$$\left(2m-7\right)\left(m+6\right)$$

Rewrite the complete factored expression.

$$2m\left(2m-7\right)\left(m+6\right)$$

Evaluate

$$2m\left(2m-7\right)\left(m+6\right)$$