Q: Let $H$ be the of subgroup $S_5$ consisting of those permutations that fix the letters 5 and 1. For each of the following pairs of elements of $S_5$, determine whether or not they belong to the same left $H$-coset. $$f_1=\begin{pmatrix}1 & 2 & 3 & 4 & 5\\3 & 2 & 5 & 4 &1\end{pmatrix},\;g_1=\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\1 & 2 & 5 &3 & 4\end{pmatrix}$$ Do $f_1$ and $g_1$ belong to the same left $H$-coset?
My answer: $$$$ Since 5 and 1 are fixed, $H=\{(234),(243)\}$. To show that these two elements are part of the same $H$-cosets, I can show that they are of the same equivalence class (because a subgroup is partitioned by a disjoint union of equivalence classes, $g_1\sim_L f_1$, which occurs $\iff g_1f_1\in H$.
try: $g_1f_1=(453)(135)=(413)\notin H\implies g_1 \not\sim_L f_1$
Did I answer this question correctly, and furthermore, is there a better way of doing this?
The $H$ you define is a subgroup of $S_{5}$, but the elements you list for it do not form a subgroup. The correct $H$ is (isomorphic to) the group of permutations of $\{ 2, 3, 4 \}$, so it is isomorphic to $S_{3}$, of order $6$. But as you'll see in the following, to solve your exercise it is not necessary to spell out the elements of $H$.
As noted in a comment, your condition for $f_{1}, g_{1}$ to be in the same left coset is incorrect, the right one being $g_{1}^{-1} f_{1} \in H$ (or equivalently $f_{1}^{-1} g_{1} \in H$).
Now $f_{1}^{-1} g_{1} \in H$ iff $1 f_{1}^{-1} g_{1} = 1$ and $5 f_{1}^{-1} g_{1} = 5$. Just check whether these hold true.