On page 4 of Landau's mechanics he says

$L(v'^2) = L(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2)$. Expanding this expression in powers of $\epsilon$ and neglecting terms above the first order, we obtain

$L(v'^2) = L(v^2) + \frac{\partial L}{\partial v^2}2\bf{v \cdot} \epsilon$.

How do we get this? Also, what is the meaning of the next line "The second term on the right of this equation is a total time derivative only if it is a linear function of the velocity $\bf v$"

This being a total time derivative would say $ \frac{\partial L}{\partial v^2}2\bf{v \cdot} \epsilon = C(t)$. Where do we get a linear function of $\bf v$ from this?