Solve 35m^2n+5mn-30n

$$35 { m }^{ 2 } n+5mn-30n$$

Factor

$$5n\left(7m-6\right)\left(m+1\right)$$

Solution Steps

Factor out 5.

$$5\left(7m^{2}n+mn-6n\right)$$

Consider 7m^{2}n+mn-6n. Factor out n.

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

Consider 7m^{2}+m-6. Factor the expression by grouping. First, the expression needs to be rewritten as 7m^{2}+am+bm-6. To find a and b, set up a system to be solved.

$$a+b=1 ab=7\left(-6\right)=-42$$

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

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

Factor out m in 7m^{2}-6m.

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

Factor out common term 7m-6 by using distributive property.

$$\left(7m-6\right)\left(m+1\right)$$

Rewrite the complete factored expression.

$$5n\left(7m-6\right)\left(m+1\right)$$

Evaluate

$$5n\left(7m-6\right)\left(m+1\right)$$