Solve 3x^2-12x+6-2x-15x^2

$$3 x ^ { 2 } - 12 x + 6 - 2 x - 15 x ^ { 2 }$$

Evaluate

$$6-14x-12x^{2}$$

Solution Steps

Combine -12x and -2x to get -14x.

$$3x^{2}-14x+6-15x^{2}$$

Combine 3x^{2} and -15x^{2} to get -12x^{2}.

$$-12x^{2}-14x+6$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$-12x^{2}-14x+6$$

Factor out 2.

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

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.

$$a+b=-7 ab=-6\times 3=-18$$

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.

$$a=2 b=-9$$

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

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

Factor out 2x in the first and 3 in the second group.

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

Factor out common term -3x+1 by using distributive property.

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

Rewrite the complete factored expression.

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