Question

which of the following could be the next step of solving the equation?
use the steps in the table below to answer the question.

initial equation:	$14x + 4(2x - 6) = x(2x + 8)$
2nd step:	$14x + 8x - 24 = 2x^2 + 8x$
3rd step:	$22x - 24 = 2x^2 + 8x$
4th step:	$14x - 24 = 2x^2$
5th step:	$-2x^2 + 14x - 24 = 0$

Explanation:

Step1: Analyze the 5th Step Equation

The 5th step equation is $-2x^{2}+14x - 24 = 0$. A common next step could be to simplify by dividing all terms by a common factor. The greatest common factor of -2, 14, and -24 is -2.

Step2: Divide Each Term by -2

Divide each term in $-2x^{2}+14x - 24 = 0$ by -2.
For the first term: $\frac{-2x^{2}}{-2}=x^{2}$
For the second term: $\frac{14x}{-2}=-7x$
For the third term: $\frac{-24}{-2} = 12$
For the fourth term: $\frac{0}{-2}=0$
So the new equation is $x^{2}-7x + 12 = 0$. Another possible next step could be factoring the quadratic or using the quadratic formula, but simplifying by dividing by -2 is a logical next step.

Answer:

A possible next step is to divide the entire equation $-2x^{2}+14x - 24 = 0$ by -2 to get $x^{2}-7x + 12 = 0$ (or other valid operations like factoring $x^{2}-7x + 12$ as $(x - 3)(x - 4)=0$ etc., depending on the method of solving quadratic equations). If we consider simplifying by dividing by -2, the next step equation is $x^{2}-7x + 12 = 0$.