Solve (x-4)(3x+6)+(x-4)(12x+48)=0

$$( x - 4 ) ( 3 x + 6 ) + ( x - 4 ) ( 12 x + 48 ) = 0$$

Solve for x

$$x = -\frac{18}{5} = -3\frac{3}{5} = -3.6$$

$$x=4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x-4 by 3x+6 and combine like terms.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Use the distributive property to multiply x-4 by 12x+48 and combine like terms.

$$3x^{2}-6x-24+12x^{2}-192=0$$

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

$$15x^{2}-6x-24-192=0$$

Subtract 192 from -24 to get -216.

$$15x^{2}-6x-216=0$$

Divide both sides by 3.

$$5x^{2}-2x-72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=-2 ab=5\left(-72\right)=-360$$

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 -360.

$$1,-360 2,-180 3,-120 4,-90 5,-72 6,-60 8,-45 9,-40 10,-36 12,-30 15,-24 18,-20$$

Calculate the sum for each pair.

$$1-360=-359 2-180=-178 3-120=-117 4-90=-86 5-72=-67 6-60=-54 8-45=-37 9-40=-31 10-36=-26 12-30=-18 15-24=-9 18-20=-2$$

The solution is the pair that gives sum -2.

$$a=-20 b=18$$

Rewrite 5x^{2}-2x-72 as \left(5x^{2}-20x\right)+\left(18x-72\right).

$$\left(5x^{2}-20x\right)+\left(18x-72\right)$$

Factor out 5x in the first and 18 in the second group.

$$5x\left(x-4\right)+18\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(5x+18\right)$$

To find equation solutions, solve x-4=0 and 5x+18=0.

$$x=4 x=-\frac{18}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-4 by 3x+6 and combine like terms.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Use the distributive property to multiply x-4 by 12x+48 and combine like terms.

$$3x^{2}-6x-24+12x^{2}-192=0$$

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

$$15x^{2}-6x-24-192=0$$

Subtract 192 from -24 to get -216.

$$15x^{2}-6x-216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -6 for b, and -216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 15\left(-216\right)}}{2\times 15}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 15\left(-216\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-6\right)±\sqrt{36-60\left(-216\right)}}{2\times 15}$$

Multiply -60 times -216.

$$x=\frac{-\left(-6\right)±\sqrt{36+12960}}{2\times 15}$$

Add 36 to 12960.

$$x=\frac{-\left(-6\right)±\sqrt{12996}}{2\times 15}$$

Take the square root of 12996.

$$x=\frac{-\left(-6\right)±114}{2\times 15}$$

The opposite of -6 is 6.

$$x=\frac{6±114}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{6±114}{30}$$

Now solve the equation x=\frac{6±114}{30} when ± is plus. Add 6 to 114.

$$x=\frac{120}{30}$$

Divide 120 by 30.

$$x=4$$

Now solve the equation x=\frac{6±114}{30} when ± is minus. Subtract 114 from 6.

$$x=-\frac{108}{30}$$

Reduce the fraction \frac{-108}{30} to lowest terms by extracting and canceling out 6.

$$x=-\frac{18}{5}$$

The equation is now solved.

$$x=4 x=-\frac{18}{5}$$

Steps for Completing the Square

Use the distributive property to multiply x-4 by 3x+6 and combine like terms.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Use the distributive property to multiply x-4 by 12x+48 and combine like terms.

$$3x^{2}-6x-24+12x^{2}-192=0$$

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

$$15x^{2}-6x-24-192=0$$

Subtract 192 from -24 to get -216.

$$15x^{2}-6x-216=0$$

Add 216 to both sides. Anything plus zero gives itself.

$$15x^{2}-6x=216$$

Divide both sides by 15.

$$\frac{15x^{2}-6x}{15}=\frac{216}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\left(-\frac{6}{15}\right)x=\frac{216}{15}$$

Reduce the fraction \frac{-6}{15} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{2}{5}x=\frac{216}{15}$$

Reduce the fraction \frac{216}{15} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{2}{5}x=\frac{72}{5}$$

Divide -\frac{2}{5}, the coefficient of the x term, by 2 to get -\frac{1}{5}. Then add the square of -\frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{72}{5}+\left(-\frac{1}{5}\right)^{2}$$

Square -\frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{72}{5}+\frac{1}{25}$$

Add \frac{72}{5} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{361}{25}$$

Factor x^{2}-\frac{2}{5}x+\frac{1}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{361}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{361}{25}}$$

Simplify.

$$x-\frac{1}{5}=\frac{19}{5} x-\frac{1}{5}=-\frac{19}{5}$$

Add \frac{1}{5} to both sides of the equation.

$$x=4 x=-\frac{18}{5}$$