First, we need to distribute the first set of parentheses:

(4x+7)(x^2+3) = 4x(x^2) + 4x(3) + 7(x^2) + 7(3) = 4x^3 + 12x + 7x^2 + 21

Next, we need to distribute the second set of parentheses:

-4/3x(x-2)(x+2) = -4/3x(x^2 - 4) = -4/3x^3 + 16/3x

Finally, we can combine like terms and simplify:

4x^3 + 7x^2 + 12x + 21 - 4/3x^3 + 16/3x - 3x^2
= (4 - 4/3)x^3 + (7 - 3)x^2 + (12 + 16/3)x + 21
= 8/3x^3 + 4x^2 + 44/3x + 21

Therefore, the simplified expression is 8/3x^3 + 4x^2 + 44/3x + 21.