Solve 30x^{3}y^2-35x^2y^2-15xy^2

$$30 x ^ { 3 } y ^ { 2 } - 35 x ^ { 2 } y ^ { 2 } - 15 x y ^ { 2 }$$

Factor

$$5x\left(2x-3\right)\left(3x+1\right)y^{2}$$

Solution Steps

Factor out 5.

$$5\left(6x^{3}y^{2}-7x^{2}y^{2}-3xy^{2}\right)$$

Consider 6x^{3}y^{2}-7x^{2}y^{2}-3xy^{2}. Factor out xy^{2}.

$$xy^{2}\left(6x^{2}-7x-3\right)$$

Consider 6x^{2}-7x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-7 ab=6\left(-3\right)=-18$$

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 -18.

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

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -7.

$$a=-9 b=2$$

Rewrite 6x^{2}-7x-3 as \left(6x^{2}-9x\right)+\left(2x-3\right).

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

Factor out 3x in 6x^{2}-9x.

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

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

$$\left(2x-3\right)\left(3x+1\right)$$

Rewrite the complete factored expression.

$$5xy^{2}\left(2x-3\right)\left(3x+1\right)$$

Evaluate

$$5x\left(2x-3\right)\left(3x+1\right)y^{2}$$