I have the following Legendre Polynomials,
$$P_{0}\left(x\right) = 1$$ $$P_{1}\left(x\right) = x$$ $$P_{2}\left(x\right) = \dfrac{1}{2}\left(3x^{2}-1\right)$$ $$P_{3}\left(x\right) = \dfrac{1}{2}\left(5x^{3}-3x\right)$$ $$P_{4}\left(x\right) = \dfrac{1}{8}\left(35x^{4}-30x^{2}+3\right)$$ $$P_{5}\left(x\right) = \dfrac{1}{8}\left(63x^{5}-70x^{3}+15x\right)$$
Is there a method, that isn't just by inspection, for expressing $f\left(x\right)=x^{3}-5x-2$ as a linear combination of the above Legendre Polynomials?
Yes. Legendre polynomials are orthogonal so you can express the coefficient in terms of an integral
$$f(x)=\sum_i \alpha_i P_i(x) \\ \alpha_i = \frac{2 i + 1}{2}\int_{-1}^1 P_i(x) f(x) dx$$