"Find smallest natural value of $n$ for which $5n+16$ and $8n +29$ have a common factor $>1$. What are the possible common factors of $5n+16$ and $8n +29$? For each possible common factor of $5n+16$ and $8n +29$ find the general form of $n$ that yields the pair of numbers with that common factor."
I tried to do this manually. I really got nowhere. I noticed that $n$ has to be a tad bit bigger than $5$... :)
Any common factor of $5n+16$ and $8n+29$ is a factor of $$5(8n+29)-8(5n+16)=17\ .$$ Since you don't want a common factor $1$, it has to be $17$. So you are looking for $$17\mid 5n+16\ ,$$ and this is easy to solve by congruence methods. I'll leave the rest up to you.
BTW, re your title - this is not the GCF of two polynomials, because $n$ is being thought of as a number. If you were thinking of $n$ as a variable with no specific value, then it would be a GCF of polynomials.