Are Recurrence Relations Related?

Since $x_1,x_2,x_3,x_4$ are roots of the given equation. Therefore: $x_1^4+2x_1^3+3x_1^2+4x_1+5=0\color{#3D99F6}{....(1)}$ $x_2^4+2x_2^3+3x_2^2+4x_2+5=0\color{#3D99F6}{....(2)}$ $x_3^4+2x_3^3+3x_3^2+4x_3+5=0\color{#3D99F6}{....(3)}$ $x_4^4+2x_4^3+3x_4^2+4x_4+5=0\color{#3D99F6}{....(4)}$

Multiplying $\color{#3D99F6}{(1)}$ by $x^n$ , $\color{#3D99F6}{(2)}$ by $2x^n$ , $\color{#3D99F6}{(3)}$ by $3x^n$ and $\color{#3D99F6}{(4)}$ by $4x^n$ and adding all gives :

$\sum_{k=1}^{4}kx_k^{n+4}+2\sum_{k=1}^{4}kx_k^{n+3}+3\sum_{k=1}^{4}kx_k^{n+2}+4\sum_{k=1}^{4}kx_k^{n+1}+5\sum_{k=1}^{4}kx_k^{n}=0$

$\implies a_{n+4}+2a_{n+3}+3a_{n+2}+4a_{n+1}+5a_n=0$ $\implies \dfrac{a_{n+4}+3a_{n+2}+5a_n}{a_{n+3}+2a_{n+1}}=-2$ Put $n=2014$ .

$\implies \dfrac{a_{2018}+3a_{2016}+5a_{2014}}{a_{2017}+2a_{2015}}=\boxed{-2}$