Suppose $f(x)=x^{2} p(x)$ for some unknown function $p(x)$.
If $p(2)=4$ and $p^{\prime}(2)=7$, then
\[
f^{\prime}(2)=
\]

Step 1 ：Given that $f(x)=x^{2} p(x)$, we can find the derivative of $f(x)$ using the product rule.

Step 2 ：The product rule states that $(uv)' = u'v + uv'$, where $u$ and $v$ are functions of $x$.

Step 3 ：In this case, $u=x^{2}$ and $v=p(x)$.

Step 4 ：So, $u'=2x$ and $v'=p'(x)$.

Step 5 ：Substituting these into the product rule gives $f'(x) = 2x p(x) + x^{2} p'(x)$.

Step 6 ：We are asked to find $f'(2)$, so we substitute $x=2$ into the equation to get $f'(2) = 2*2*p(2) + 2^{2}*p'(2)$.

Step 7 ：Given that $p(2)=4$ and $p'(2)=7$, we substitute these values into the equation to get $f'(2) = 2*2*4 + 2^{2}*7$.

Step 8 ：Solving this gives $f'(2) = 16 + 28$.

Step 9 ：So, $f'(2) = \boxed{44}$.