Let's factor the given expression [tex]\(x^2 + 4x - 12\)[/tex].
We are dealing with a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -12\)[/tex].
To factor this, we need to find two numbers that multiply to [tex]\(a \cdot c = 1 \cdot (-12) = -12\)[/tex] and add up to [tex]\(b = 4\)[/tex].
Step-by-step process:
1. Write down the pairs of factors for [tex]\(-12\)[/tex]:
- [tex]\((1, -12)\)[/tex]
- [tex]\((-1, 12)\)[/tex]
- [tex]\((2, -6)\)[/tex]
- [tex]\((-2, 6)\)[/tex]
- [tex]\((3, -4)\)[/tex]
- [tex]\((-3, 4)\)[/tex]
2. Identify the pair that adds up to [tex]\(4\)[/tex]. We see that:
- [tex]\((-2) + 6 = 4\)[/tex]
Therefore, [tex]\(-2\)[/tex] and [tex]\(6\)[/tex] are the correct numbers.
3. Rewrite the middle term ([tex]\(4x\)[/tex]) using [tex]\(-2\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[
x^2 + 4x - 12 = x^2 - 2x + 6x - 12
\][/tex]
4. Group the terms:
[tex]\[
(x^2 - 2x) + (6x - 12)
\][/tex]
5. Factor out the common factors in each group:
[tex]\[
x(x - 2) + 6(x - 2)
\][/tex]
6. Factor by grouping:
[tex]\[
(x + 6)(x - 2)
\][/tex]
Thus, the factored form of the expression [tex]\(x^2 + 4x - 12\)[/tex] is [tex]\((x - 2)(x + 6)\)[/tex].
The correct factorization is:
[tex]\[
(x - 2)(x + 6)
\][/tex]
