16h^2+8hk-15k^2 is homogeneous: all the terms are of order 2. As a result, factoring this polynomial is similar to the problem of factoring the polynomial 16x^2+8x-15, which is of the form ax^2+bx+c, with a=16, b=8 and c=-15.
This has discriminant given by the formula:
Delta = b^2-4ac
= 8^2-(4xx16xx-15) = 64+960 = 1024 = 32^2
So 16x^2+8x-15=0 has roots given by:
x = (-b+-sqrt(Delta))/(2a) = (-8+-32)/32 = (-1+-4)/4
That is x = -5/4 or x = 3/4.
Multiplying these two equations through by the denominators, we can deduce that (4x+5) and (4x-3) are factors:
Hence 16x^2+8x-15 = (4x+5)(4x-3)
If we substitute h/k for x, this becomes:
16(h/k)^2+8(h/k)-15 = (4(h/k)+5)(4(h/k)-3)
Now multiply both sides by k^2 to find
16h^2+8hk-15k^2 = (4h+5k)(4h-3k)
