Solve 30x^3y-9x^2y-12xy

$$30 x ^ { 3 } y - 9 x ^ { 2 } y - 12 x y$$

Factor

$$3xy\left(5x-4\right)\left(2x+1\right)$$

Solution Steps

Factor out 3.

$$3\left(10x^{3}y-3x^{2}y-4xy\right)$$

Consider 10x^{3}y-3x^{2}y-4xy. Factor out xy.

$$xy\left(10x^{2}-3x-4\right)$$

Consider 10x^{2}-3x-4. Factor the expression by grouping. First, the expression needs to be rewritten as 10x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-3 ab=10\left(-4\right)=-40$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -3.

$$a=-8 b=5$$

Rewrite 10x^{2}-3x-4 as \left(10x^{2}-8x\right)+\left(5x-4\right).

$$\left(10x^{2}-8x\right)+\left(5x-4\right)$$

Factor out 2x in 10x^{2}-8x.

$$2x\left(5x-4\right)+5x-4$$

Factor out common term 5x-4 by using distributive property.

$$\left(5x-4\right)\left(2x+1\right)$$

Rewrite the complete factored expression.

$$3xy\left(5x-4\right)\left(2x+1\right)$$

Evaluate

$$3xy\left(5x-4\right)\left(2x+1\right)$$