To determine which trinomial is factored using algebra tiles, we need to analyze the given trinomials and examine which can be factored into two binomials.
Considering the options:
1. [tex]\(x^2 + 3x - 6\)[/tex]
2. [tex]\(x^2 + 5x - 6\)[/tex]
3. [tex]\(x^2 + 3x - 2\)[/tex]
4. [tex]\(x^2 + x - 6\)[/tex]
We need to find the correct pair of binomials whose product results in each of the given trinomials. Let’s proceed step by step:
### Step-by-step Factorization:
#### 1. [tex]\(x^2 + 3x - 6\)[/tex]
To factor this trinomial, we need two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(3\)[/tex] (the coefficient of the linear term).
Considerations for pairs:
- [tex]\(1 \cdot -6\)[/tex]
- [tex]\(-1\cdot 6\)[/tex]
- [tex]\(2 \cdot -3\)[/tex]
- [tex]\(-2 \cdot 3\)[/tex]
Not any of these pairs add to [tex]\(3\)[/tex].
#### 2. [tex]\(x^2 + 5x - 6\)[/tex]
To factor this trinomial, we need to find two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(5\)[/tex].
Considerations for pairs:
- [tex]\(1 \cdot -6\)[/tex]
- [tex]\(-1 \cdot 6\)[/tex]
- [tex]\(2 \cdot -3\)[/tex]
- [tex]\(-2 \cdot 3\)[/tex]
Again, none of these pairs add to [tex]\(5\)[/tex].
#### 3. [tex]\(x^2 + 3x - 2\)[/tex]
We need to find two numbers that multiply to [tex]\(-2\)[/tex] and add to [tex]\(3\)[/tex].
Considerations for pairs:
- [tex]\(1 \cdot -2\)[/tex]
- [tex]\(-1 \cdot 2\)[/tex]
Once more, the pairs don’t add up to [tex]\(3\)[/tex].
#### 4. [tex]\(x^2 + x - 6\)[/tex]
Here, we need two numbers that multiply to [tex]\(-6\)[/tex] and add up to [tex]\(1\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
Considerations for pairs:
- [tex]\(1 \cdot -6\)[/tex]
- [tex]\(-1 \cdot 6\)[/tex]
- [tex]\(2 \cdot -3\)[/tex]
- [tex]\(-2 \cdot 3\)[/tex]
Upon review:
- [tex]\(3 \cdot -2 = -6\)[/tex]
- [tex]\(3 + (-2) = 1\)[/tex]
This pair fits perfectly!
So, the trinomial [tex]\(x^2 + x - 6\)[/tex] can be factored into [tex]\((x + 3)(x - 2)\)[/tex].
Thus, the correct trinomial which is factored as described is:
[tex]\(\boxed{x^2 + x - 6}\)[/tex]
