To factor the expression \( \frac{1}{4}x + \frac{15}{4} \) completely, we can start by factoring out the common factor from the two terms.
Both terms share a common factor of \( \frac{1}{4} \). We can factor this out:
\[ \frac{1}{4}(x + 15) \]
Now we can look for the options given:
A. \( \frac{1}{2}( \frac{1}{2}x + \frac{15}{4}) \)
B. \( \frac{1}{4}(x + \frac{7}{2}) \)
C. \( \frac{1}{4}(x + 15) \)
D. \( \frac{3}{4}(x + 3) \)
The factored expression we found is:
\[ \frac{1}{4}(x + 15) \]
Thus, the correct choice is:
C. \(\frac{1}{4}(x + 15)\).