Martin solved the quadratic equation 3x

Question

2 + 2x − 8 = 0. His work is shown below. However, he
accidentally spilled ink over the step where he divided by the GCF that he found for each pair.
Which of the following options shows the missing part of Martin’s work?

x(3x + 6) + 4(x + 2) = 0
3x(x + 2) − 4(x + 2) = 0
3x(x + 2) + 4(x + 2) = 0
3x(x + 2) − 4(x − 2) = 0

Answer 1

To solve the quadratic equation \(3x^2 + 2x - 8 = 0\), Martin likely factored it by grouping.

The equation can be grouped as follows:

1. First, we can rearrange it: \(3x^2 + 2x - 8 = 0\).

2. The terms can be grouped based on a common factor. Notice that the quadratic can be expressed in a way to facilitate factoring. We can look for two pairs:

   Consider the first line: \(3x^2 + 6x - 4x - 8\) which can be grouped in pairs: \( (3x^2 + 6x) + (-4x - 8) \)

3. Now, factor out the GCF from each pair: \(3x(x + 2) - 4(x + 2) = 0\)

4. At this point, notice that both terms contain \((x + 2)\), thus you can factor that out: \((3x - 4)(x + 2) = 0\)

Now that we have reviewed the factoring steps, looking at your options, the missing step where Martin divided or factored out the greatest common factor is:

Option: \(3x(x + 2) - 4(x + 2) = 0\)

This is the correct step showing the necessary factoring before proceeding to the next step of solving the equation.