Martin solved the quadratic equation 3x2+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?

Question

Several lines of equations.(1 point)
Responses

3x(x+2)−4(x−2)=0
3 x left parenthesis x plus 2 right parenthesis minus 4 left parenthesis x minus 2 right parenthesis equals 0

3x(x+2)+4(x+2)=0
3 x left parenthesis x plus 2 right parenthesis plus 4 left parenthesis x plus 2 right parenthesis equals 0

3x(x+2)−4(x+2)=0
3 x left parenthesis x plus 2 right parenthesis minus 4 left parenthesis x plus 2 right parenthesis equals 0

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

Answer 1

To solve the quadratic equation \( 3x^2 + 2x - 8 = 0 \) using factoring by grouping, let's first rewrite it:

1. We need to factor by grouping. We recognize that there are two pairs of terms in the expression. First, we derive the grouping based on the terms in the equation.

2. We can find the GCF (Greatest Common Factor). In this case, we can organize as:

   \( 3x^2 + 6x - 4x - 8 = 0 \)

   Splitting \( 2x \) into \( 6x \) and \( -4x \) allows us to factor it by grouping.

3. Now grouping the terms: \((3x^2 + 6x) + (-4x - 8) = 0\)

4. Taking out the GCF from each group: \[ 3x(x + 2) - 4(x + 2) = 0 \]

This implies that Martin has the correct factorization. Upon analyzing the provided options:

• The first option is: \[ 3x(x+2)−4(x−2)=0 \] (Incorrect)

• The second option is: \[ 3x(x+2)+4(x+2)=0 \] (Incorrect)

• The third option is: \[ 3x(x+2)−4(x+2)=0 \] (Correct)

• The fourth option is: \[ x(3x+6)+4(x+2)=0 \] (Incorrect)

Therefore, the missing part of Martin's work is:

\[ 3x(x+2)−4(x+2)=0 \] (option 3)