Let's walk through the detailed process of factoring [tex]\(3x^2 - 12\)[/tex].
### Step 1: Factor out the Greatest Common Factor (GCF)
First, identify the GCF of the terms in the expression [tex]\(3x^2\)[/tex] and [tex]\(-12\)[/tex]. The GCF is [tex]\(3\)[/tex].
So, we can factor out [tex]\(3\)[/tex]:
[tex]\[
3x^2 - 12 = 3(x^2 - 4)
\][/tex]
### Step 2: Check if Further Factoring is Possible
Next, we observe the expression inside the parentheses [tex]\(x^2 - 4\)[/tex]. This expression is a difference of squares, which can be factored further using the identity:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
Here, [tex]\(x^2 - 4\)[/tex] fits the form [tex]\(a^2 - b^2\)[/tex] where [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex]. Therefore,
[tex]\[
x^2 - 4 = (x + 2)(x - 2)
\][/tex]
### Step 3: Write the Completely Factored Form
Substituting this back into our factored form:
[tex]\[
3(x^2 - 4) = 3(x + 2)(x - 2)
\][/tex]
### Conclusion
The expression [tex]\(3(x^2 - 4)\)[/tex] is equivalent to [tex]\(3x^2 - 12\)[/tex], but it is not completely factored because [tex]\(x^2 - 4\)[/tex] can still be factored further into [tex]\((x + 2)(x - 2)\)[/tex].
Thus, the correct statement regarding [tex]\(3(x^2 - 4)\)[/tex] is:
The expression is equivalent, but it is not completely factored.
So, the answer is:
[tex]\[
\text{The expression is equivalent, but it is not completely factored.}
\][/tex]
