= Solution
The decisive observation is that $x$ is itself an <eigenfunction>:
$$
\widehat Qx=\frac d{dx}\left[(x^2-1)\frac{dx}{dx}\right]=2x.
$$
On the regular <Legendre polynomial> domain, integration by parts has endpoint factor $x^2-1$, so the boundary form vanishes for regular functions at $x=\pm1$. The interval alone is not an endpoint condition; regularity is the usual condition selecting the <Legendre polynomials>. Their <eigenvalues> are $n(n+1)$, and the $n=1$ normalized function is
$$
\psi_1(x)=\sqrt{\frac32}\,x,\qquad \int_{-1}^{1}|\psi_1|^2dx=1.
$$
Therefore
$$
\boxed{x=\sqrt{\frac23}\,\psi_1(x),\qquad c_1=\sqrt{\frac23},\quad c_n=0\ (n\ne1).}
$$
Equivalently, the <Hermitian operator> identity gives $(q_n-2)c_n=0$, and nondegeneracy leaves only one possible nonzero coefficient. The unnormalized function $x$ has squared <norm> $2/3$, consistent with this coefficient.
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