2. Let two acute angles $\alpha, \beta$ satisfy
$$
\begin{array}{l}
(\sin \alpha+\cos \alpha)(\sin \beta+\cos \beta)=2 \text {. } \\
\text { Then }(\sin 2 \alpha+\cos 3 \beta)^{2}+(\sin 2 \beta+\cos 3 \alpha)^{2} \\
=
\end{array}
$$
$2.3-2 \sqrt{2}$.
From the given, we know
$$
\begin{array}{l}
\sin (\alpha+\beta)+\cos (\alpha-\beta)=2 \\
\Rightarrow \sin (\alpha+\beta)=\cos (\alpha-\beta)=1 \\
\Rightarrow \alpha+\beta=90^{\circ}, \alpha=\beta \\
\Rightarrow \alpha=\beta=45^{\circ} .
\end{array}
$$

Then $(\sin 2 \alpha+\cos 3 \beta)^{2}+(\sin 2 \beta+\cos 3 \alpha)^{2}$
$$
=2\left(\sin 90^{\circ}+\cos 135^{\circ}\right)^{2}=3-2 \sqrt{2} .
$$