Set the given quadratic polynomial equal to zero to find its zeroes.

The common term is $4u$.

$$4u(u + 2) = 0$$

Set each factor equal to zero and solve for $u$.

$$4u = 0 \quad \text{or} \quad u + 2 = 0$$

$$u = \frac{0}{4} \quad \text{or} \quad u = -2$$ $$u = 0 \quad \text{or} \quad u = -2$$

The zeroes of the polynomial $4u^2 + 8u$ are 0 and $-2$.

The final answer is $\text{0, -2}$.