Solve 6x^2y^2-10xy^2-84y^2

$$6 x ^ { 2 } y ^ { 2 } - 10 x y ^ { 2 } - 84 y ^ { 2 }$$

Factor

$$2\left(3x-14\right)\left(x+3\right)y^{2}$$

Solution Steps

Factor out 2.

$$2\left(3x^{2}y^{2}-5xy^{2}-42y^{2}\right)$$

Consider 3x^{2}y^{2}-5xy^{2}-42y^{2}. Factor out y^{2}.

$$y^{2}\left(3x^{2}-5x-42\right)$$

Consider 3x^{2}-5x-42. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-42. To find a and b, set up a system to be solved.

$$a+b=-5 ab=3\left(-42\right)=-126$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -126.

$$1,-126 2,-63 3,-42 6,-21 7,-18 9,-14$$

Calculate the sum for each pair.

$$1-126=-125 2-63=-61 3-42=-39 6-21=-15 7-18=-11 9-14=-5$$

The solution is the pair that gives sum -5.

$$a=-14 b=9$$

Rewrite 3x^{2}-5x-42 as \left(3x^{2}-14x\right)+\left(9x-42\right).

$$\left(3x^{2}-14x\right)+\left(9x-42\right)$$

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

$$x\left(3x-14\right)+3\left(3x-14\right)$$

Factor out common term 3x-14 by using distributive property.

$$\left(3x-14\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$2y^{2}\left(3x-14\right)\left(x+3\right)$$

Evaluate

$$2\left(3x-14\right)\left(x+3\right)y^{2}$$