Take the part you highlighted:
Also, forgive me if I'm wrong but you wrote: A = arcsin[4sin34/5] ≈ 26.574° then The only other possibilities are (180 - A) = 153.426
But you didn't highlight:
....but 153.426 + 43 > 180, so this would not be a triangle.
Otherwise, take a look at c) again, as it is one of the data set which describes two possible triangles.
c) A = arcsin[4sin34/2.237] ≈ 89.181 and 90.819
Therefore two triangles can be formed.
Just work it out;
If sinA = asinB/b; sinA = 4sin34/2.237
and A = arcsin(asinB/b), then A = 89.181 is one possiblility
but...180 - 89.181 = 90.819° is another possibility,
Since 34 + 90.819 < 180, this angle is possible,
Well, 4sin34/2.237 = .99989...
and sin90.819 = .99989...
and sin89.789 = .99989...
This all just means that you can form two different triangles meeting all of the criteria listed in the question.
That seemed long, but I hope it explained it to you. It is just that two different angles can produce the same length side.