= Solution
For a compact space $X$, $K^0(X)$ is the <Grothendieck group> of isomorphism classes of finite-rank complex <vector bundles> under direct sum. Tensor product descends to this group and makes it a ring, with $[\mathbb C_X]$ as its unit.
The hypothesis $E_0\oplus\mathbb C_X\cong E_1\oplus\mathbb C_X$ says that $E_0$ and $E_1$ define the same stable class. Rank-$d$ bundles are classified by maps to $BU(d)$, and the stabilization $BU(d)\to BU$ is $2d$-connected. Since the finite <CW complex> $X$ has dimension $k\leq2d$, stabilization is injective on $[X,BU(d)]$. Thus <stable cancellation for complex vector bundles> gives
$$
\boxed{E_0\cong E_1.}
$$
Now let $E_0,E_1$ have rank $d$ over <Complex projective space> $\mathbb{CP}^d$ and have the same <Chern classes>. Their <Chern characters> agree because each component of $\operatorname{ch}(E)$ is a universal rational polynomial in the Chern classes. The ring
$$
K^0(\mathbb{CP}^d)\cong\mathbb Z[t]/(t^{d+1})
$$
is torsion-free, while the Chern character becomes an isomorphism after tensoring with $\mathbb Q$; it is therefore injective. Hence $[E_0]=[E_1]$ in K-theory, so after adding trivial bundles they are isomorphic. Since $\mathbb{CP}^d$ has real dimension $2d$, cancellation applies once more:
$$
\boxed{c(E_0)=c(E_1)\Longrightarrow E_0\cong E_1.}
$$
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