= Solution
At $P_1=(0,0)$ the tangent slope is $-2$, so its equation is $y=-2x$. Substitution leaves $x^3=0$; this tangent meets the curve three times at $P_1$. The <chord-and-tangent group law> therefore gives
$$
\boxed{2P_1=(0,-1)=-P_1,\qquad3P_1=O.}
$$
In particular $P_1$ has exact order three. The chord through $P_1$ and $P_2=(-2,-4)$ is $y=2x$. Substitution gives $x(x+2)(x-2)=0$, so the third intersection is $(2,4)$. Negating it gives
$$
\boxed{P_1\oplus P_2=(2,-5).}
$$
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