Sure, let's factor the given quadratic expression step by step.
Given expression:
[tex]\[ 5x^2 - 45 \][/tex]
### Step 1: Factor out the greatest common factor (GCF)
First, we notice that both terms [tex]\(5x^2\)[/tex] and [tex]\(-45\)[/tex] have a common factor of 5. So, we can factor out 5 from the expression:
[tex]\[ 5(x^2 - 9) \][/tex]
### Step 2: Recognize the difference of squares
Next, we observe that the expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. Recall the difference of squares formula:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
In this case, [tex]\(x^2 - 9\)[/tex] can be written as [tex]\(x^2 - 3^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex].
### Step 3: Apply the difference of squares formula
So, we apply the formula:
[tex]\[ x^2 - 9 = (x + 3)(x - 3) \][/tex]
### Step 4: Substitute back into the factored form
We now substitute this result back into our expression:
[tex]\[ 5(x^2 - 9) = 5(x + 3)(x - 3) \][/tex]
Thus, the completely factored form of the expression [tex]\(5x^2 - 45\)[/tex] is:
[tex]\[ 5(x + 3)(x - 3) \][/tex]
### Conclusion
Looking at the options provided:
A. [tex]\(5(x + 3)(x - 3)\)[/tex] <== This matches our solution.
B. [tex]\(5(x - 9)(x + 9)\)[/tex]
C. [tex]\(5(x^2 - 45)\)[/tex]
D. [tex]\((5x + 9)(x - 5)\)[/tex]
The correct option is:
[tex]\[ \boxed{A} \][/tex]
