Solve 5s^2y-25sy+20y

$$5 s ^ { 2 } y - 25 s y + 20 y$$

Factor

$$5y\left(s-4\right)\left(s-1\right)$$

Solution Steps

Factor out 5.

$$5\left(s^{2}y-5sy+4y\right)$$

Consider s^{2}y-5sy+4y. Factor out y.

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

Consider s^{2}-5s+4. Factor the expression by grouping. First, the expression needs to be rewritten as s^{2}+as+bs+4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

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

$$\left(s^{2}-4s\right)+\left(-s+4\right)$$

Factor out s in the first and -1 in the second group.

$$s\left(s-4\right)-\left(s-4\right)$$

Factor out common term s-4 by using distributive property.

$$\left(s-4\right)\left(s-1\right)$$

Rewrite the complete factored expression.

$$5y\left(s-4\right)\left(s-1\right)$$

Evaluate

$$5y\left(s-4\right)\left(s-1\right)$$