To factor the quadratic expression 6x² + 5x - 6, we need to find two binomials that, when multiplied together, give us the original expression. The factoring process requires us to find a pair of numbers that both add to the coefficient of the middle term (which is 5 in this case) and multiply to the product of the coefficient of the x² term and the constant term (which is 6 * -6 = -36).
The first step is to list the pairs of factors that multiply to -36:
-1 and 36
1 and -36
-2 and 18
2 and -18
-3 and 12
3 and -12
-4 and 9
4 and -9
-6 and 6
Now, we need to check these pairs to see which combine to give us 5 when added. It turns out that 3 and -2 add up to 1 and when we multiply 3 by 2, we obtain the necessary -36 when considering the 6x² term`s coefficient. To factor this using the numbers 3 and -2, we must consider the 6x² coefficient as well. In other words, we split the middle term, 5x, into two terms that will use our numbers 3 and -2.
The pair of factors from the list above that add up to 5 (our middle coefficient) considering the 6 in 6x² are 3 and 2 with appropriate multipliers: (2 * 3) and (3 * -2) which equal 6 and -6, respectively. This allows us to rewrite 5x as 6x - x.
Now our expression looks like this: 6x² + 6x - x - 6.
We then group the terms to factor by grouping:
(6x² + 6x) - (x + 6)
Factoring 6x out of the first group gives us:
6x * (x + 1) - (x + 6)
There's no common factor in the second group, but we notice that if we take -1 out of the second group, we can rearrange the expression so that it mirrors the first group:
6x * (x + 1) - 1 * (x + 6)
Now we see that the binomial (x + 1) can't be our factor since it doesn't help us reach a solution.
Getting back to our factoring pair 3 and -2, we aim to represent the number 3 as a multiple of the term with x, which would be (2x + 3) since 3 is 3/2 times 2, and to get "-2" we tweak it further.
Doing this on the original expression, we rewrite 5x like this: 3 (2x) + (-2) 3, reformulating the expression:
6x² + 3(2x) + (-2)(3) - 6
Group terms to form common factors:
(6x² + 3(2x)) + ((-2)(3) - 6)
Factor out common terms:
3(2x² + (2x)) - 2((3) + 3)
Now, we have the common binomial (2x + 3):
(2x + 3)(3x - 2)
Checking the signs and coefficients, the factored form of 6x² + 5x - 6 is indeed:
(2x + 3)(3x - 2)
From the provided options, the correct answer is (2x + 3) and (3x - 2).
