To find the product of two complex numbers \( (a, b) \) and \( (c, d) \), where \( a \) and \( b \) are the real and imaginary parts of the first complex number, and \( c \) and \( d \) are the real and imaginary parts of the second complex number, we use the formula:

\[
(ac - bd, ad + bc)
\]

For the given complex numbers \( (4, 3) \) and \( (5, -6) \):

- \( a = 4 \), \( b = 3 \)
- \( c = 5 \), \( d = -6 \)

Now, calculate:

1. The real part:
\[
ac - bd = (4 \times 5) - (3 \times -6) = 20 + 18 = 38
\]

2. The imaginary part:
\[
ad + bc = (4 \times -6) + (3 \times 5) = -24 + 15 = -9
\]

So the product is \( (38, -9) \).

The correct answer is:

D. (38, -9).