Represent the given signal in mathematical form
Let the horizontal axis by time t and vertical axis by signal value x(t).

From t=-3 to t=3 the first part you wrote
$$ x(t) = -2 r(t+3) + 2 r(t) + 16 u(t) - r(t) - 2 u(t-3) $$
is correct.

From t=0 to t=3, the signal x(t) above is sloping down with \$\frac {dx(t)}{dt} = \frac{7-10}{3-0} = -1\$.
According to given graph, this slope should become \$\frac {dx(t)}{dt} = \frac{0-5}{5-3}= -2.5\$, so, a ramp giving a slope of -3/2 should be added.
$$ x(t) = \bigg( -2 r(t+3) + 2 r(t) + 16 u(t) - r(t) - 2 u(t-3) \bigg) - \frac{3}{2} r(t-3) $$

To compensate after t=5, the slope of -2.5 should be nullified:
$$ x(t) = \bigg( -2 r(t+3) + 2 r(t) + 16 u(t) - r(t) - 2 u(t-3) - \frac{3}{2} r(t-3) \bigg) + \frac{5}{2} r(t-5)$$