To determine the step where Martin divided by the GCF, let's analyze the given quadratic equation and the factoring step. The original equation is:
\[ 3x^2 + 2x - 8 = 0 \]
Next, we need to factor the left-hand side of the equation. The goal in factoring a quadratic expression \( ax^2 + bx + c \) is to express it as a product of two binomials. In Martin's work, he factored out a common expression from the first two terms and from the last two terms. The expression we need to find matches the way he factored out the GCF from each part.
Looking at the provided options:
- Option 1: \( 3x(x+2)−4(x+2)=0 \)
- Option 2: \( 3x(x+2)−4(x−2)=0 \)
- Option 3: \( x(3x+6)+4(x+2)=0 \)
- Option 4: \( 3x(x+2)+4(x+2)=0 \)
Since Martin originally worked with \( 3x^2 + 2x - 8 = 0 \), we want to separate \( 3x^2 + 2x \) and \( -8 \) into a factored expression.
The correct way to factor this expression would involve identifying pairs:
- The first part \( 3x^2 + 2x \) can have \( (x+2) \) factored out if we apply decomposition correctly with their respective coefficients.
- The second part confirms that there should be a matching \( (x+2) \) as a factor.
Thus, with the matching terms, we realize that the common factor here is indeed \( (x + 2) \).
Looking at the options again, Option 1 shows both parts successfully combining to lead to a common factor:
So, the correct choice that fits the requirements of Martin's step of factoring by grouping would be:
Option 1: \( 3x(x+2)−4(x+2)=0 \)
1 answer