To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with a domain of \( (0, \infty) \), we can express \( p(x) \) in terms of \( h(x) \).
Since we are looking for a square root function, we can take the square root of \( h(x) \):
\[ p(x) = \sqrt{h(x)} = \sqrt{4x^2} \]
Simplifying this, we get:
\[ p(x) = \sqrt{4} \cdot \sqrt{x^2} = 2x \]
So, we can express \( p(x) \) in terms of the form \( \sqrt{ax} \).
To match the form \( p(x) = \sqrt{ax} \), notice that \( p(x) = 2x \) can be rewritten as:
\[ p(x) = \sqrt{(2^2)(x^2)} = \sqrt{4x^2} \]
In this case, \( a = 4 \).
Therefore, the answer is:
\[ a = 4 \]