Convexity of $f(x) = \log \left( \sum_{i = 1}^N \varepsilon_i\left( Q(a_i + b_i\sqrt x ) \right) ^2 \right)$

Question

I know that for $x,a,b\ge0$; ${Q({a_i} + {b_i}\sqrt x )}^2$ holds convexity because the $Q$ function holds the convexity (monotonicity) . I dont know how should I approach to prove

$$\log \left( \sum_{i = 1}^N \varepsilon_i\left( Q(a_i + b_i\sqrt x ) \right) ^2 \right)$$

is convex for $x \ge 0$.