Textbook version:
R = \frac{v_0^2 \sin(2\theta)}{g}
Simple, neat, and completely unrealistic.
Reality: add air resistance with
F_d = kv^2
and suddenly you’re knee-deep in differential equations. The range drops, the best angle is no longer 45°, and your model just got real.
Exam trap: forgetting to resolve the initial velocity components.
u_x = u \cos\theta, \qquad u_y = u \sin\theta
That’s two marks gone before you even blink.