To complete the square for the expression \( q^2 + 20q \), we need to find the number that makes it a perfect-square quadratic.
- Take the coefficient of \( q \), which is 20.
- Divide it by 2: \( \frac{20}{2} = 10 \).
- Square that result: \( 10^2 = 100 \).
Thus, the number we need to add to \( q^2 + 20q \) to complete the square is \( 100 \).
Therefore, we rewrite the expression as:
\[ q^2 + 20q + 100 \]
This simplifies to:
\[ (q + 10)^2 \]
So, the number that makes the polynomial a perfect-square quadratic is \( \boxed{100} \).