can someone please help me check my work for this question? I want to make sure I did it correctly.
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That's correct, but it would be easier and faster to use
(i) $v=u^3=CL^2$ instead of $u$ as your utility function (which is allowed since $v$ is a positive transformation of $u$), and
(ii) the rule that in the optimum the ratio of marginal utilities equals the ratio of prices. This immediately gives you $7.5L=2C$, and substituting $2C$ for the $7.5L$ in the budget constraint then yields the solution.