Joining Two Pieces Without a Crease

A charging model is built from two formulas stitched together at $x = 2$:

$f(x) = \begin{cases} a x^{2} + b x, & x \le 2 \\[4pt] 5x - 3, & x > 2 \end{cases}$

The constants $a$ and $b$ have not been chosen yet. They must be chosen so that $f$ is differentiable at $x = 2$ — the curve has to arrive at the join with no gap and no corner.

The right-hand piece is a straight line, so its slope is simply the coefficient of $x$. The left-hand piece's slope at $x = 2$ has to come from the limit definition

$f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$

applied to $a x^{2} + b x$.

Find the value of $a + b$.

Round your answer to two decimal places.