Given that $$(m+n+p)(mn+mp+np)=25$$ and that $$m^2(n+p)+n^2(m+p)+p^2(m+n)=4$$ for real numbers $m$, $n$, and $p$, what is the value of $mnp$?
Expanding the first given equation using the distributive property, we have \begin{align*}
25&=(m+n+p)(mn+mp+np)\\
&=m\cdot(mn+mp+np)+n\cdot(mn+mp+np)\\
&\qquad+p\cdot(mn+mp+np)\\
&=m^2n+m^2p+mnp+mn^2+mnp\\
&\qquad +n^2p+mnp+mp^2+np^2\\
&=3mnp+m^2n+m^2p+mn^2+n^2p+mp^2+np^2
\end{align*} Expanding the second given equation using the distributive property, we have \begin{align*}
4&=m^2(n+p)+n^2(m+p)+p^2(m+n)\\
&=m^2n+m^2p+mn^2+n^2p+mp^2+np^2\end{align*} We substitute the equation $$4=m^2n+m^2p+mn^2+n^2p+mp^2+np^2$$ into the expanded form of the first given equation to get \[25=3mnp+4\] or $mnp=7$.