we have that
m∠B = 45° b = 4 c = 5
we know that
from the Law of Sines, obtain
sin C/c=sin B/b
sin C=c*sin B/b-------> 5*sin 45/4---------> 0.884
C=arc sin(0.884)---------> C=62.1°
A+B+C=180°
so
m∠A = 180 - 45 - 62.1 = 72.9°
a=(sin A/sin B)*b--------> (sin 72.9/sin 45)*4------> 5.41
The first triangle has∠A=72.9°, m∠B=45°, m∠C = 62.1°, a=5.41, b=4, c=5.
Also,
C=arc sin(0.884)--------> 180-62.1--------> C=117.9°
A+B+C=180°
som∠A = 180 - 45 - 117.9 = 17.1°
a=(sin A/sin B)*b--------> (sin 17.1/sin 45)*4------> 1.66
The second triangle has
m∠A = 17.1°, m∠B = 45°, m∠C = 117.9°, a = 1.66, b = 4, c = 5
The answer is
the number of possible triangles are 2