To solve the equation \( \frac{5}{8} + \frac{1}{4}(2x) = \frac{5}{4} \), follow these steps:

1. Simplify the term inside the parentheses:
\[
\frac{1}{4}(2x) = \frac{2x}{4} = \frac{x}{2}
\]

2. Substitute this back into the equation:
\[
\frac{5}{8} + \frac{x}{2} = \frac{5}{4}
\]

3. To eliminate the fractions, find a common denominator. The common denominator for 8, 2, and 4 is 8. Rewrite each term with the common denominator:
\[
\frac{5}{8} + \frac{4x}{8} = \frac{10}{8}
\]

4. Combine the terms on the left side:
\[
\frac{5 + 4x}{8} = \frac{10}{8}
\]

5. Since the denominators are the same, you can equate the numerators:
\[
5 + 4x = 10
\]

6. Solve for \( x \):
\[
4x = 10 - 5
\]
\[
4x = 5
\]
\[
x = \frac{5}{4}
\]

So, the solution to the equation is:
\[
x = \frac{5}{4}
\]