To find the square of the expression \( (2q - 1)^2 \), we can use the formula for the square of a binomial, which states that \( (a - b)^2 = a^2 - 2ab + b^2 \).
In this case, let \( a = 2q \) and \( b = 1 \).
Now, we apply the formula:
\[ (2q - 1)^2 = (2q)^2 - 2(2q)(1) + (1)^2 \]
Calculating each term:
- \( (2q)^2 = 4q^2 \)
- \( -2(2q)(1) = -4q \)
- \( (1)^2 = 1 \)
Now, we can combine these results:
\[ (2q - 1)^2 = 4q^2 - 4q + 1 \]
So the simplified answer is:
\[ \boxed{4q^2 - 4q + 1} \]