To simplify the expression, we can start by expanding both terms.

First we expand (d+4)^3 using the binomial theorem:
(d+4)^3 = d^3 + 3d^2(4) + 3d(4)^2 + (4)^3
= d^3 + 12d^2 + 48d + 64

Next we expand (d+1)(d+3):
(d+1)(d+3) = d^2 + 3d + d + 3
= d^2 + 4d + 3

Now we can substitute these back into the original expression:
(d+4)^3 - (d+1)(d+3)
= (d^3 + 12d^2 + 48d + 64) - (d^2 + 4d + 3)
= d^3 + 12d^2 + 48d + 64 - d^2 - 4d - 3
= d^3 + 11d^2 + 44d + 61

Therefore, (d+4)^3 - (d+1)(d+3) simplifies to d^3 + 11d^2 + 44d + 61.