In (a), $M(t)$ is a polynomial, so there is some large $k$ such that the $2k$th derivative of $M$ is zero. But then $E[X^{2k}] = M^{(2k)}(t) = 0$. Since $X^{2k} \ge 0$, the only way it can have expectation equal to zero is if $P(X^{2k} = 0)=1$. But then $P(X=0)=1$.

In (a), $M(t)$ is a polynomial, so there is some large $k$ such that the $2k$th derivative of $M$ is zero. But then $E[X^{2k}] = M^{(2k)}(0) = 0$. Since $X^{2k} \ge 0$, the only way it can have expectation equal to zero is if $P(X^{2k} = 0)=1$. But then $P(X=0)=1$.