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.