Solve 6y+11xy+3x^2y

$$6 y + 11 x y + 3 x ^ { 2 } y$$

Factor

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

Solution Steps

Factor out y.

$$y\left(6+11x+3x^{2}\right)$$

Consider 6+11x+3x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3x^{2}+11x+6$$

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=11 ab=3\times 6=18$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=2 b=9$$

Rewrite 3x^{2}+11x+6 as \left(3x^{2}+2x\right)+\left(9x+6\right).

$$\left(3x^{2}+2x\right)+\left(9x+6\right)$$

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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