To determine which multiplication expression is equivalent to [tex]\(\frac{x+8}{x^2} + \frac{2x+16}{2x^2}\)[/tex], let's start by simplifying the original expression step by step.
1. Consider the original expression:
[tex]\[
\frac{x+8}{x^2} + \frac{2x+16}{2x^2}
\][/tex]
2. Factor the numerator in the second fraction:
[tex]\[
\frac{2x+16}{2x^2} = \frac{2(x+8)}{2x^2} = \frac{x+8}{x^2}
\][/tex]
3. The expression now is:
[tex]\[
\frac{x+8}{x^2} + \frac{x+8}{x^2}
\][/tex]
4. Combine the two fractions:
[tex]\[
\frac{(x+8) + (x+8)}{x^2} = \frac{2(x+8)}{x^2}
\][/tex]
This further simplifies to:
[tex]\[
2 \cdot \frac{x+8}{x^2}
\][/tex]
5. Next, we want to find the equivalent multiplication expression. We analyze the given options:
a. [tex]\(\frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
b. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2}\)[/tex]
c. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
6. For each of these options, let's simplify to see if it matches [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex]:
- For the first option:
[tex]\[
\frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16}
\][/tex]
Simplifying the second fraction:
[tex]\[
\frac{2 x^2}{2(x+8)} = \frac{x^2}{x+8}
\][/tex]
Then, multiplying both fractions:
[tex]\[
\frac{x^2}{x+8} \cdot \frac{x^2}{x+8} = \left(\frac{x^2}{x+8}\right)^2
\][/tex]
This does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the second option:
[tex]\[
\frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2}
\][/tex]
Simplifying the second fraction:
[tex]\[
\frac{2(x+8)}{2x^2} = \frac{x+8}{x^2}
\][/tex]
Then, multiplying both fractions:
[tex]\[
\frac{x+8}{x^2} \cdot \frac{x+8}{x^2} = \left(\frac{x+8}{x^2}\right)^2
\][/tex]
This also does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the third option:
[tex]\[
\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}
\][/tex]
Simplifying the second fraction:
[tex]\[
\frac{2 x^2}{2(x+8)} = x^2 \cdot \frac{1}{(x+8)}
\][/tex]
Then, multiplying both fractions:
[tex]\[
\frac{x+8}{x^2} \cdot \frac{x^2}{x+8} = 1
\][/tex]
Hence:
[tex]\[
2 \cdot \frac{x+8}{x^2}
\][/tex]
This matches the original simplified expression.
Therefore, the correct multiplication expression is:
[tex]\[
\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}
\][/tex]
The correct choice is:
[tex]\[
\boxed{\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}}
\][/tex]
