Solve 12ky^2+8ky-20k

$$12 k y ^ { 2 } + 8 k y - 20 k$$

Factor

$$4k\left(y-1\right)\left(3y+5\right)$$

Solution Steps

Factor out 4.

$$4\left(3ky^{2}+2ky-5k\right)$$

Consider 3ky^{2}+2ky-5k. Factor out k.

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

Consider 3y^{2}+2y-5. Factor the expression by grouping. First, the expression needs to be rewritten as 3y^{2}+ay+by-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 3y^{2}+2y-5 as \left(3y^{2}-3y\right)+\left(5y-5\right).

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

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

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

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

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

Rewrite the complete factored expression.

$$4k\left(y-1\right)\left(3y+5\right)$$

Evaluate

$$4k\left(y-1\right)\left(3y+5\right)$$