Can both
n + 3 and
n^2 + 3 be perfect cubes if n is an integer ?
If both (n+3) and (n^2+3) were perfect cubes, the product (n+3)(n^2+3) = n^3+3n^2+3n+9 = (n+1)^3 + 8 = (n+1)^3 + 2^3 will be a perfect cube, right. Now one can see geometrically that if we take any cube formed by n^3 smaller cubic units, we can't construct a greater cube by adding 8 cubic units to the original cube, for any n. Thus the assumption is false.
Solution following Jason's
