In Fulton and Harris's Text Representation Theory: A First Course, exercise 1.12(b) asks to show that $\operatorname{Sym}^{k+6}V \cong \operatorname{Sym}^kV \oplus R$ as representations of $\frak S_3$. $V$ is the 2-dimensional standard representation of $\frak S_3$ and $R$ is the regular representation. Note that this is supposed to be done without using character theory.
The hint given by my professor was to show that \operatorname{Sym}^6 V \cong U \oplus R \cong U^{\oplus 2} \oplus U' \oplus V^{\oplus 2}, where $U$ is the trivial representation and U' is the alternating representation. Then, we're supposed to use that to find copies of $\operatorname{Sym}^k V$ and $R$ in $\operatorname{Sym}^{k+6} V$ that intersect only at $0$, which will then prove the original isomorphism.
Proving the congruence was relatively straightforward, but I have no idea how we're supposed to find copies of $\operatorname{Sym}^k V$ and $R$ in $\operatorname{Sym}^{k+6} V$.