To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with the domain of \( (0, \infty) \), we start by expressing \( h(x) \) in terms of \( p(x) \).
Given \( h(x) = 4x^2 \), we can think of \( p(x) \) such that it relates to \( h(x) \) through \( p(x) \). A square root function related to the given quadratic would involve taking the square root of \( h(x) \).
First, we rewrite the quadratic function:
\[ h(x) = 4x^2 \]
Now, taking the square root of both sides, we have:
\[ \sqrt{h(x)} = \sqrt{4x^2} \]
This simplifies to:
\[ \sqrt{h(x)} = 2x \]
Since we want \( p(x) \) to be in the form \( p(x) = a \sqrt{x} \) for some constant \( a \), we note that we want to express \( p(x) \) in terms of \( x \) rather than \( 2x \).
To express it in the form \( p(x) = a \sqrt{x} \):
Setting \( x = \frac{p(x)}{2} \), we can form a function that represents it when \( h(x) \) is squared back:
\[ p(x) = 2\sqrt{x} \]
Upon comparing with the form \( p(x) = a \sqrt{x} \), we see that \( a = 2 \).
Thus, the answer is:
\[ a = 2 \]
Therefore, the related square root function is:
\[ p(x) = 2\sqrt{x} \]