Solve 5x^2y+10xy-30x^3y

$$5 x ^ { 2 } y + 10 x y - 30 x ^ { 3 } y$$

Factor

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

Solution Steps

Factor out 5.

$$5\left(x^{2}y+2xy-6x^{3}y\right)$$

Consider x^{2}y+2xy-6x^{3}y. Factor out xy.

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

Consider x+2-6x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-6x^{2}+x+2$$

Factor the expression by grouping. First, the expression needs to be rewritten as -6x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=1 ab=-6\times 2=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=4 b=-3$$

Rewrite -6x^{2}+x+2 as \left(-6x^{2}+4x\right)+\left(-3x+2\right).

$$\left(-6x^{2}+4x\right)+\left(-3x+2\right)$$

Factor out 2x in -6x^{2}+4x.

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

Factor out common term -3x+2 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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