To determine which of the given expressions is equivalent to [tex]\(\frac{8p + 8}{-32p + 4}\)[/tex], we need to simplify the given fraction step by step.
1. Simplify the numerator and the denominator separately:
The numerator is [tex]\(8p + 8\)[/tex]. We can factor out an 8:
[tex]\[
8p + 8 = 8(p + 1)
\][/tex]
The denominator is [tex]\(-32p + 4\)[/tex]. We can factor out a 4:
[tex]\[
-32p + 4 = 4(-8p + 1)
\][/tex]
2. Rewrite the expression using factored forms:
[tex]\[
\frac{8(p + 1)}{4(-8p + 1)}
\][/tex]
3. Simplify the fraction by dividing the numerator and the denominator by 4:
[tex]\[
\frac{8(p + 1)}{4(-8p + 1)} = \frac{8}{4} \cdot \frac{p + 1}{-8p + 1} = 2 \cdot \frac{p + 1}{-8p + 1}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{2(p + 1)}{-8p + 1}
\][/tex]
Now we compare this result with the given options:
(A) [tex]\(\frac{p + 2}{-4p + 1}\)[/tex]
(B) [tex]\(\frac{2p + 2}{-8p + 1}\)[/tex]
(C) [tex]\(\frac{2p + 2}{8p + 1}\)[/tex]
(D) [tex]\(p + 1\)[/tex]
The equivalent expression to [tex]\(\frac{2(p + 1)}{-8p + 1}\)[/tex] is
(B) [tex]\(\frac{2p + 2}{-8p + 1}\)[/tex].
Therefore, the correct choice is [tex]\(\boxed{B}\)[/tex].
