Solve 50x^2y+5xy-10y

$$50 x ^ { 2 } y + 5 x y - 10 y$$

Factor

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

Solution Steps

Factor out 5.

$$5\left(10x^{2}y+xy-2y\right)$$

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

$$y\left(10x^{2}+x-2\right)$$

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

$$a+b=1 ab=10\left(-2\right)=-20$$

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 -20.

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

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 1.

$$a=-4 b=5$$

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

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

Factor out 2x in 10x^{2}-4x.

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

Factor out common term 5x-2 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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