To simplify the expression \( 7 \sqrt{21} + 4 \sqrt{84} \), we can start by simplifying \( \sqrt{84} \):
\[ \sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2 \sqrt{21} \]
Now we can substitute \( \sqrt{84} \) back into the original expression:
\[ 4 \sqrt{84} = 4 \times 2 \sqrt{21} = 8 \sqrt{21} \]
Now, we can combine the terms:
\[ 7 \sqrt{21} + 8 \sqrt{21} = (7 + 8) \sqrt{21} = 15 \sqrt{21} \]
Thus, the simplified expression is:
\[ \boxed{15 \sqrt{21}} \]