Solve 4x^3y-44x^2y+112xy

$$4 x ^ { 3 } y - 44 x ^ { 2 } y + 112 x y$$

Factor

$$4xy\left(x-7\right)\left(x-4\right)$$

Solution Steps

Factor out 4.

$$4\left(x^{3}y-11x^{2}y+28xy\right)$$

Consider x^{3}y-11x^{2}y+28xy. Factor out xy.

$$xy\left(x^{2}-11x+28\right)$$

Consider x^{2}-11x+28. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=-11 ab=1\times 28=28$$

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 28.

$$-1,-28 -2,-14 -4,-7$$

Calculate the sum for each pair.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

The solution is the pair that gives sum -11.

$$a=-7 b=-4$$

Rewrite x^{2}-11x+28 as \left(x^{2}-7x\right)+\left(-4x+28\right).

$$\left(x^{2}-7x\right)+\left(-4x+28\right)$$

Factor out x in the first and -4 in the second group.

$$x\left(x-7\right)-4\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x-4\right)$$

Rewrite the complete factored expression.

$$4xy\left(x-7\right)\left(x-4\right)$$

Evaluate

$$4xy\left(x-7\right)\left(x-4\right)$$