= Solution
Use the standard <sl2 triple> $H=x\partial_x-y\partial_y$, $X=x\partial_y$, $Y=y\partial_x$. In the <tensor product of Lie algebra representations>, each operator acts on both factors. Put
$$
u_{ij}=x^{3-i}y^i\otimes x^{2-j}y^j,\quad0\leq i\leq3,\quad0\leq j\leq2.
$$
These twelve vectors form a <weight basis>, and $Hu_{ij}=(5-2(i+j))u_{ij}$. Thus the bases of the <weight spaces> are
$$
\begin{array}{c|l}
\text{weight}&\text{basis}\\\hline
5&u_{00}\\
3&u_{10},u_{01}\\
1&u_{20},u_{11},u_{02}\\
-1&u_{30},u_{21},u_{12}\\
-3&u_{31},u_{22}\\
-5&u_{32}
\end{array}
$$
There are no other weights. The <weight diagram> shows their multiplicities, with the decomposition also displayed for comparison.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-1-sl2-weights.png]
{title=Weights and multiplicities of the cubic by quadratic sl2 tensor product and its irreducible summands}
The <raising operator> acts by
$$
Xu_{ij}=i\,u_{i-1,j}+j\,u_{i,j-1},
$$
with out-of-range terms omitted. At weights five, three and one its kernels are respectively spanned by
$$
\boxed{h_5=u_{00},\qquad h_3=u_{10}-u_{01},\qquad h_1=u_{20}-2u_{11}+u_{02}.}
$$
At weight one the kernel equations are $2a+b=0$ and $b+2c=0$ for $au_{20}+bu_{11}+cu_{02}$. At weight minus one they are $3a+b=0$, $2b+2c=0$, $c=0$, which force zero; the maps on the two remaining negative-weight spaces are injective as well. Thus these three lines give \b[all nonzero <highest-weight vectors> up to scalar], at their respective weights. A linear combination of different lines is killed by $X$ but is not a <weight vector>, so is not a <highest-weight vector>.
The <lowering operator> is
$$
Yu_{ij}=(3-i)u_{i+1,j}+(2-j)u_{i,j+1}.
$$
In particular,
$$
\boxed{h_1=u_{20}-2u_{11}+u_{02},\qquad Yh_1=u_{30}-2u_{21}+u_{12}}
$$
is an explicit <basis> for the submodule $\Gamma_1$. Indeed $Y^2h_1=0$, $Xh_1=0$, $X(Yh_1)=h_1$, and the two vectors have weights $1,-1$. The analogous strings from $h_5,h_3$ have dimensions six and four. Their distinct irreducible highest weights and total dimension twelve give the <sl2 tensor product of cubic and quadratic symmetric powers>
$$
\boxed{U\cong\Gamma_5\oplus\Gamma_3\oplus\Gamma_1.}
$$
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