Solve 6n^3y+4n^2y-10ny

$$6 n ^ { 3 } y + 4 n ^ { 2 } y - 10 n y$$

Factor

$$2ny\left(n-1\right)\left(3n+5\right)$$

Solution Steps

Factor out 2.

$$2\left(3n^{3}y+2n^{2}y-5ny\right)$$

Consider 3n^{3}y+2n^{2}y-5ny. Factor out ny.

$$ny\left(3n^{2}+2n-5\right)$$

Consider 3n^{2}+2n-5. Factor the expression by grouping. First, the expression needs to be rewritten as 3n^{2}+an+bn-5. To find a and b, set up a system to be solved.

$$a+b=2 ab=3\left(-5\right)=-15$$

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 -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.

$$a=-3 b=5$$

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

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

Factor out 3n in the first and 5 in the second group.

$$3n\left(n-1\right)+5\left(n-1\right)$$

Factor out common term n-1 by using distributive property.

$$\left(n-1\right)\left(3n+5\right)$$

Rewrite the complete factored expression.

$$2ny\left(n-1\right)\left(3n+5\right)$$

Evaluate

$$2ny\left(n-1\right)\left(3n+5\right)$$