Step 1 :The given expression is \(\frac{3}{v+2}\) and we need to find the equivalent expression with the denominator \((v+5)(v+2)\).
Step 2 :To find the equivalent expression, we can multiply the numerator and denominator of the given expression by \((v+5)\).
Step 3 :This gives us the equivalent expression as \(3*(v + 5)\), which simplifies to \(3v + 15\).
Step 4 :This is the numerator of the equivalent rational expression with the denominator \((v+5)(v+2)\).
Step 5 :Final Answer: The equivalent rational expression is \(\frac{3v + 15}{(v+5)(v+2)}\). So, the blank should be filled with \(\boxed{3v + 15}\).