= Solution
A pure bipartite state is <entangled> when it is not a <product state> $|u\rangle_A\otimes|v\rangle_B$.
Since
$$
H\otimes H|11\rangle
=\frac12(|00\rangle-|01\rangle-|10\rangle+|11\rangle),
$$
we have
$$
|\alpha\rangle
=\frac1{2\sqrt3}
(3|00\rangle+|01\rangle+|10\rangle-|11\rangle).
$$
Its coefficient matrix is proportional to
$$
\begin{pmatrix}3&1\\1&-1\end{pmatrix},
$$
whose determinant is $-4\ne0$. By the <two-qubit product-state determinant criterion>, a product state would have determinant zero. \b[Therefore $|\alpha\rangle$ is entangled.]
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