Answer
$B=\color{blue}{62.7^\circ}$ or $B=\color{blue}{117.3^\circ}$
Work Step by Step
Using the Sine Law,
$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} \\
\dfrac{\sin 45.6^\circ}{456} = \dfrac{\sin B}{567}\\
\sin B = \dfrac{567 \sin 45.6^\circ}{456} \\
\sin B \approx 0.8884 \\
\color{blue}{B \approx 62.7^\circ}\quad \text{or}\quad \color{blue}{B\approx 180^\circ-62.7^\circ \;=\; 117.3^\circ}
$
If $B=62.7^\circ$, then, since $A+B+C=180^\circ$,
$C = 180^\circ-A-B \\
C = 180^\circ-45.6^\circ-62.7^\circ \\
C = 71.7^\circ$.
Thus, such a triangle is possible.
If $B=117.3^\circ$, then, since $A+B+C=180^\circ$,
$C = 180^\circ-A-B \\
C = 180^\circ-45.6^\circ-117.3^\circ \\
C = 17.1^\circ$.
Thus, such a triangle is also possible.
Thus, there are exactly two solutions: $\color{blue}{B=62.7^\circ}$ or $\color{blue}{B=117.3^\circ}$.
