To solve the problem of finding the factors of [tex]\(4b^2 - 16\)[/tex], we can follow these steps:
1. Identify the given expression:
The expression to be factored is [tex]\(4b^2 - 16\)[/tex].
2. Factor out the greatest common factor (GCF):
Both terms [tex]\(4b^2\)[/tex] and [tex]\(-16\)[/tex] are divisible by 4. Thus, we factor out 4:
[tex]\[
4b^2 - 16 = 4(b^2 - 4)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
The expression [tex]\(b^2 - 4\)[/tex] is a difference of squares, which can be factored using the formula [tex]\(a^2 - b^2 = (a-b)(a+b)\)[/tex]. Here, [tex]\(b^2 - 4\)[/tex] is of the form [tex]\(a^2 - b^2\)[/tex] with [tex]\(a = b\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
b^2 - 4 = (b - 2)(b + 2)
\][/tex]
4. Combine the factors:
Substitute back the factors into the original factored expression:
[tex]\[
4(b^2 - 4) = 4(b - 2)(b + 2)
\][/tex]
Thus, the factors of [tex]\(4b^2 - 16\)[/tex] are [tex]\(4(b - 2)(b + 2)\)[/tex].
Given the options:
A. [tex]\((2b - 2)(2b + 4)\)[/tex]
B. [tex]\((2b + 4)(2b + 4)\)[/tex]
C. [tex]\(2(b - 4)(b + 4)\)[/tex]
D. [tex]\(4(b - 2)(b + 2)\)[/tex]
The correct answer is D. [tex]\(4(b - 2)(b + 2)\)[/tex].
