Show that P is closed against the Kleene star

I have that question that looks kinda easy at first but it is quite hard.


Let $L\in P$. Prove that $L^*\in P$

my approach:

I tried to generate a Turing machine but I got stuck with the thing of the poly time machine, I only managed to create a non-deterministic machine. I know that there is a complicated method with graphs or with dynamic programming, But I am asking for a full proof or a more gentle one, "computational models"-oriented and not with algorithmic approach, but both kinds of proof will be good.

Hint: Use dynamic programming. If the input is $x_1 \ldots x_n$, compute inductively whether $x_1 \ldots x_i \in L^*$. Use the fact that you can check whether $x_{j+1} \ldots x_i \in L$ in polynomial time.

StackExchange Computer Science Q#59382, answer score: 11