= Solution
For $\mathfrak{sl}_2$, the Verma module has basis
$$
v,,fv,,f^2v,\ldots
$$
by the <Poincare-Birkhoff-Witt theorem>. If $hv=\lambda v$, then
$$
e f^rv=r(\lambda-r+1)f^{r-1}v.
$$
For $\lambda=-d$ with $d>0$, the coefficient is
$$
r(-d-r+1)\ne0
$$
for every $r\geq1$. Thus no $f^rv$ with $r>0$ is a <singular vector>.
Any nonzero submodule contains a weight vector $f^rv$ because the $h$-weights are distinct. Applying $e^r$ gives a nonzero multiple of $v$, after which applying powers of $f$ generates all of $M(-d)$. Hence
$$
\boxed{M(-d)\text{ is irreducible and infinite-dimensional}.}
$$
This is also the negative-highest-weight case of <Reducibility of an sl2 Verma module>.
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