Let's go through the process of factoring the trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex] step-by-step.
1. Identify the trinomial: The expression given is [tex]\(6x^2 - xy - 2y^2\)[/tex].
2. Look for a common factor: In this case, there are no common factors among all the terms, so we proceed to factor by grouping or using other techniques.
3. Factoring the trinomial:
- We need to find two binomials that, when multiplied together, give the original trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex].
- Notice we need to find two numbers that multiply to the product of the leading coefficient (6) and the constant term (-2), which is [tex]\(6 \times (-2) = -12\)[/tex], and add up to the middle coefficient (-1).
4. Finding the pairs:
- The pairs that multiply to [tex]\(-12\)[/tex] are: [tex]\( (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), (-3, 4) \)[/tex].
- Among these pairs, [tex]\((-3, 4)\)[/tex] adds up to the middle coefficient [tex]\(-1\)[/tex].
5. Rewrite the middle term:
- We rewrite [tex]\(6x^2 - xy - 2y^2\)[/tex] using [tex]\(-3xy + 4xy\)[/tex]:
[tex]\[6x^2 - 3xy + 4xy - 2y^2\][/tex]
6. Group the terms:
- Group the terms to factor by grouping:
[tex]\[6x^2 - 3xy + 4xy - 2y^2 = (6x^2 - 3xy) + (4xy - 2y^2)\][/tex]
7. Factor each group:
- From [tex]\(6x^2 - 3xy\)[/tex], factor out the common term [tex]\(3x\)[/tex]:
[tex]\[3x(2x - y)\][/tex]
- From [tex]\(4xy - 2y^2\)[/tex], factor out the common term [tex]\(2y\)[/tex]:
[tex]\[2y(2x - y)\][/tex]
8. Combine the factors:
- We get:
[tex]\[3x(2x - y) + 2y(2x - y)\][/tex]
- Notice that [tex]\((2x - y)\)[/tex] is a common term:
[tex]\[(3x + 2y)(2x - y)\][/tex]
So, the factored form of the trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex] is:
[tex]\[
6x^2 - xy - 2y^2 = (3x + 2y)(2x - y)
\][/tex]
Therefore, the correct choice is:
[tex]\[
\text{A. } 6x^2 - xy - 2y^2 = (3x + 2y)(2x - y)
\][/tex]
