Solve 6x^4y+15x^3y-9x^2y

$$6 x ^ { 4 } y + 15 x ^ { 3 } y - 9 x ^ { 2 } y$$

Factor

$$3y\left(2x-1\right)\left(x+3\right)x^{2}$$

Solution Steps

Factor out 3.

$$3\left(2x^{4}y+5x^{3}y-3x^{2}y\right)$$

Consider 2x^{4}y+5x^{3}y-3x^{2}y. Factor out x^{2}y.

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

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

$$a+b=5 ab=2\left(-3\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=-1 b=6$$

Rewrite 2x^{2}+5x-3 as \left(2x^{2}-x\right)+\left(6x-3\right).

$$\left(2x^{2}-x\right)+\left(6x-3\right)$$

Factor out x in the first and 3 in the second group.

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

Factor out common term 2x-1 by using distributive property.

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

Rewrite the complete factored expression.

$$3x^{2}y\left(2x-1\right)\left(x+3\right)$$

Evaluate

$$3y\left(2x-1\right)\left(x+3\right)x^{2}$$