= Solution
Here
$$
4a^3+27b^2=4(-9)^3+27(9)^2=-729=-3^6.
$$
By the <Lutz–Nagell theorem>, a nonzero rational torsion point has integral coordinates and either $y=0$ or $y^2\mid3^6$. Part b gives $v_3(y)\leq1$, so $y\in\{0,\pm1,\pm3\}$. The cubic $x^3-9x+9$ has no integral zero. Substitution of $y=\pm1,\pm3$ gives exactly
$$
\pm P,\quad \pm Q,\quad \pm(P+Q),\quad \pm(P-Q).
$$
This proves the required inclusion.
Since $2Q=-Q$, the point $Q$ has order three. The point $P$ is not torsion because $2P=(9/4,3/8)$ is nonintegral, contradicting Nagell–Lutz. If $P+Q$ or $P-Q$ were torsion, adding the torsion point $\mp Q$ would make $P$ torsion; their negatives are excluded in the same way. Hence
$$
E(\mathbb Q)_{\mathrm{tors}}=\{O,Q,-Q\}\cong\mathbb Z/3\mathbb Z.
$$
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