To determine the correct grouping that allows us to factor the quadratic expression [tex]\( 6x^2 + 7x - 5 \)[/tex], we need to find the correct pair of middle terms that sum to [tex]\( 7x \)[/tex] and whose product equals the product of the coefficient of [tex]\( x^2 \)[/tex] and the constant term. In this case, the coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 6 \)[/tex], and the constant term is [tex]\(-5\)[/tex]. Therefore, the correct middle terms should come from pairs of terms that multiply to [tex]\( 6 \times (-5) = -30 \)[/tex] and sum to [tex]\( 7x \)[/tex].
Let's evaluate each option step-by-step:
### Option 1: [tex]\( 6x^2 - 4x + 11x - 5 \)[/tex]
- Sum of middle terms:
[tex]\[
-4x + 11x = 7x
\][/tex]
- Product of middle terms:
[tex]\[
(-4) \times 11 = -44 \quad (\text{not equal to } -30)
\][/tex]
This option does not work.
### Option 2: [tex]\( 6x^2 - 2x + 9x - 5 \)[/tex]
- Sum of middle terms:
[tex]\[
-2x + 9x = 7x
\][/tex]
- Product of middle terms:
[tex]\[
(-2) \times 9 = -18 \quad (\text{not equal to } -30)
\][/tex]
This option does not work.
### Option 3: [tex]\( 6x^2 - 6x + 9x - 9 \)[/tex]
- Sum of middle terms:
[tex]\[
-6x + 9x = 3x \quad (\text{not equal to } 7x)
\][/tex]
This option does not work.
### Option 4: [tex]\( 6x^2 - 3x + 10x - 5 \)[/tex]
- Sum of middle terms:
[tex]\[
-3x + 10x = 7x
\][/tex]
- Product of middle terms:
[tex]\[
(-3) \times 10 = -30 \quad (\text{equal to } -30)
\][/tex]
This option works because both the sum and product conditions are satisfied.
Thus, the correct grouping of the quadratic [tex]\( 6x^2 + 7x - 5 \)[/tex] is given by:
[tex]\[
6x^2 - 3x + 10x - 5
\][/tex]
The correct answer is:
[tex]\[ \boxed{6x^2 - 3x + 10x - 5} \][/tex]
