Consider the rational function $\dfrac{f(x)}{g(x)} \in \mathbb{F}_{n}[x]$ such that $g(x) = p(x) \cdot h(x)$. If $\gcd\left(p(x), h(x)\right) = 1$, then $$\dfrac{f(x)}{g(x)} = \dfrac{f(x)}{p(x) \cdot h(x)} = t(x) + \dfrac{r(x)}{p(x)} + \dfrac{s(x)}{h(x)}$$ where $t(x), r(x), s(x) \in \mathbb{F}_{n}[x]$ and $\deg(r(x)) < \deg(p(x))$, $\deg(s(x)) < \deg(h(x))$.
Let's say, for example, that I want to decompose the rational function $\dfrac{3x^{2} + 5}{\left(3x^{2} + 2x - 3\right)\left(x^{6} + 3x^{5} + 4x^{2} - 3x + 2\right)}$ over $\mathbb{F}_{7}[x]$ into partial fractions, I know that $$\dfrac{3x^{2} + 5}{\left(3x^{2} + 2x - 3\right)\left(x^{6} + 3x^{5} + 4x^{2} - 3x + 2\right)} \equiv t(x) + \dfrac{r(x)}{\left(3x^{2} + 2x - 3\right)} + \dfrac{s(x)}{\left(x^{6} + 3x^{5} + 4x^{2} - 3x + 2\right)}$$ My question is, how do we obtain the polynomials $r(x)$ and $s(x)$? Do we assume $r(x) = ax + b$ and $s(x) = cx^{5} + dx^{4} + ex^{3} + fx^{2} + gx + h$ and solve like we do on partial fraction problems taught in preliminary classes?
Thanks.