Prove the following language is regular?
The answer depends on whether $L_1$ is a language over $\{a,b\}$ or over $\{a,b,c\}$.

$L_1$ is a language over $\{a,b\}$

In this case, the easiest way to proceed is using closure operations. Show first (by constructing a DFA) that the following language is regular:
$$
L_2 = \{wcv \mid |w|_a+2|v|_b \equiv 3 \bmod 5 , w,v \in \{a,b\}^*\}.
$$
Your language is $L = L_1cL_1 \cap L_2$.

$L_1$ is a language over $\{a,b,c\}$

In this case we have to be more careful. Given a DFA (or an NFA) for $L_1$, we construct one for $L$ which in three stages:


Simulate $L_1$ and keep track of the number of $a$s modulo 5. Whenever at an accepting state of $L_1$, add a nondeterministic move to stage 2.
Read $c$.
Simulate $L_1$ (starting from its initial state again) and keep track of the number of $b$s modulo 5. A state is accepting if it is accepting for $L_1$ and the constraint $|w|_a+2|v|_b$ is satisfied, where $w$ is the word read at the first stage, and $v$ is the word read at the third stage.


I'll let you fill in the details.