I think $3\cdot k+1\cdot l=7$ is okay, as the powers of $x$ add up to $7$. We either have degree $3$ (from $x^3$) or degree $1$ (from $x$) and that's where the $3$ and $1$ come from here. $k$ represents how many degree $3$ and $l$ represents how many degree $1$ are used.

If that's still a little confusing, the main idea is that there are two ways to get $x^7$:

Case (i): $x^7$ comes from terms of the form $(x^3)^2\cdot x^1\cdot 2^2$ (we must have $2$ powers of $2$ so we have $5$ terms in total). This has a coefficient of $\dfrac{5!}{2!\cdot 1!\cdot 2!}$.

Case (ii): $x^7$ comes from terms of the form $(x^3)^1\cdot x^4\cdot 2^0$ (here there are no powers of $2$). This has a coefficient of $\dfrac{5!}{1!\cdot 4!\cdot 0!}$.

Hence, the final $x^7$ term is $$\frac{5!}{2!\cdot 1!\cdot 2!}\cdot (x^3)^2\cdot x^1\cdot 2^2 + \frac{5!}{1!\cdot 4!\cdot 0!}\cdot (x^3)^1\cdot x^4\cdot 2^0 = (120+5)x^7 = 125x^7.$$