= Solution
Write $U_0=D_+(t_0)=\operatorname{Spec}\mathbb C[x]$, where $x=t_1/t_0$, and $U_1=D_+(t_1)=\operatorname{Spec}\mathbb C[y]$, with $y=1/x$ on the overlap. Since $\mathcal G$ is a <quasi-coherent sheaf>, it is associated to a $\mathbb C[x]$-module $M$ on $U_0$. The preimages under the open inclusion $f$ of $U_0,U_1,U_0\cap U_1$ are $U_0,D(x),D(x)$, all <affine schemes>. Thus $f$ is an <affine morphism> and its <direct image sheaf> $f_*\mathcal G$ is quasi-coherent by <direct image of a quasi-coherent sheaf under an affine morphism>.
The two-chart cover is an <acyclic cover> by <vanishing of quasi-coherent cohomology on an affine scheme>, so the <acyclic cover theorem> identifies its <Čech cohomology> with <sheaf cohomology>. Its only potentially nontrivial positive-degree differential is
$$
M\oplus M_x\longrightarrow M_x,\qquad (m,n)\longmapsto n-m/1.
$$
This is surjective because of the second summand. There are no normalized Čech terms in degree two or higher for a two-open cover, so
$$
\boxed{H^i(\mathbb P^1_{\mathbb C},f_*\mathcal G)=0\qquad(i>0).}
$$
This is the <acyclic direct image from an affine chart of the projective line>. The pushforward need not itself be a flasque sheaf; the acyclic affine cover is what justifies this computation.
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