Sure, let's go through the process of factoring the expression step-by-step to identify the correct answer.
Given expression:
[tex]\[ 5x^2 - 45 \][/tex]
### Step 1: Factor out the Greatest Common Factor (GCF)
First, we identify the greatest common factor of the terms in the expression. Here, 5 is the GCF of the terms [tex]\(5x^2\)[/tex] and [tex]\(-45\)[/tex]:
[tex]\[ 5x^2 - 45 = 5(x^2 - 9) \][/tex]
### Step 2: Recognize and apply the difference of squares
Next, we notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares. Recall that a difference of squares can be factored using the formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In this case, [tex]\(x^2 - 9\)[/tex] fits the structure of a difference of squares, where:
[tex]\[ x^2 - 9 = x^2 - 3^2 \][/tex]
Applying the difference of squares formula:
[tex]\[ x^2 - 3^2 = (x - 3)(x + 3) \][/tex]
Therefore, we further factor [tex]\(5(x^2 - 9)\)[/tex] as follows:
[tex]\[ 5(x^2 - 9) = 5(x - 3)(x + 3) \][/tex]
### Conclusion
The completely factored form of [tex]\(5x^2 - 45\)[/tex] is:
[tex]\[ 5(x - 3)(x + 3) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
Thus, the option C, [tex]\(5(x + 3)(x - 3)\)[/tex], is the correct factorization of the given expression.
