Step 1 :Find the value of the functions $h$ and $g$ at $x=4$.
Step 2 :For $g(4)$, substitute $x=4$ into the function $g(x)=3x+3$ to get $g(4)=3*4+3=15$.
Step 3 :For $h(4)$, substitute $x=4$ into the function $h(x)=-4x+1$ to get $h(4)=-4*4+1=-15$.
Step 4 :The product of the functions $h$ and $g$ evaluated at $x=4$ is $(h \cdot g)(4)=h(4)*g(4)=-15*15=-225$.
Step 5 :So, $(h \cdot g)(4)=-225$.
Step 6 :Final Answer: \(\boxed{-225}\)