Derivative of a cubic polynomial via the limit definition

Start by computing $f(x+h) = (x+h)^3 + 4(x+h) = x^3 + 3x^2h + 3xh^2 + h^3 + 4x + 4h$. Subtracting $f(x) = x^3 + 4x$ gives $f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3 + 4h$. Every term contains a factor of $h$, which we factor out: $f(x+h)-f(x) = h(3x^2 + 3xh + h^2 + 4)$.

Dividing by $h$ gives $\frac{f(x+h)-f(x)}{h} = 3x^2 + 3xh + h^2 + 4$. Taking the limit as $h \to 0$, the terms with $h$ vanish, leaving $f'(x) = 3x^2 + 4$.