1. The problem is to simplify the expression $v\dot{a^2b} + ab^2$. 2. First, clarify the notation: $v\dot{a^2b}$ likely means $v$ times the derivative of $a^2b$ with respect to some variable (say $t$). 3. Use the product rule for derivatives: if $f = a^2b$, then $$\frac{d}{dt}(a^2b) = \frac{d}{dt}(a^2) \cdot b + a^2 \cdot \frac{d}{dt}(b).$$ 4. The derivative of $a^2$ is $2a \dot{a}$, so $$\frac{d}{dt}(a^2b) = 2a \dot{a} b + a^2 \dot{b}.$$ 5. Multiply this by $v$: $$v \cdot \frac{d}{dt}(a^2b) = v(2a \dot{a} b + a^2 \dot{b}) = 2v a b \dot{a} + v a^2 \dot{b}.$$ 6. Now add the remaining term $ab^2$: $$2v a b \dot{a} + v a^2 \dot{b} + a b^2.$$ 7. This is the simplified expression combining the derivative term and the additional term. Final answer: $$2v a b \dot{a} + v a^2 \dot{b} + a b^2.$$