= Solution
By the <spectral theorem>, the nonscalar unitary V has <orthonormal> <eigenvectors> $|v_0\rangle,|v_1\rangle$ with distinct <eigenvalues> $\lambda_0\ne\lambda_1$. Choose $|b\rangle=(|v_0\rangle+|v_1\rangle)/\sqrt2$. Then $V|b\rangle=(\lambda_0|v_0\rangle+\lambda_1|v_1\rangle)/\sqrt2$ is not proportional to $|b\rangle$.
On the product input $|+\rangle\otimes|b\rangle$, the <controlled unitary gate> produces
$$
U(|+\rangle\otimes|b\rangle)=\frac1{\sqrt2}\bigl(|0\rangle\otimes|b\rangle+|1\rangle\otimes V|b\rangle\bigr).
$$
Its two conditional target <vectors> are <linearly independent>, so the <conditional-vector product-state criterion> proves that the output is entangled. Equivalently, in the control <computational basis> and target eigenbasis its <coefficient matrix> is $\tfrac12\bigl(\begin{smallmatrix}1&1\\\lambda_0&\lambda_1\end{smallmatrix}\bigr)$, whose determinant is $(\lambda_1-\lambda_0)/4\ne0$. Thus \b[U is an <entangling gate>]. The excluded scalar case would only apply a <phase gate> to the control and could not entangle the factors; this is the <controlled-unitary entangling criterion>.
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