= Solution
Massive young stars emit ionizing ultraviolet radiation that creates <H II regions>. Recombination and radiative cascades produce <H-alpha> emission, including the $n=3\to2$ transition. Shocks or other ionizing sources can also produce it, but disk star-forming gas is the natural interpretation for an extended spiral-galaxy rotation measurement.
The systemic <redshift> and inclination of an intrinsically circular thin disk are
$$
\boxed{z=\frac{689.1-656.3}{656.3}\simeq0.04998,\qquad
\cos i=\frac{7.5}{30}=\frac14,\quad i\simeq75.52^\circ.}
$$
The rotation-induced half-width is $(689.7-688.5)/2=0.6\,\mathrm{nm}$. Divide by the systemic observed wavelength to remove the common redshift factor before applying the small-velocity <Doppler shift>:
$$
v_{\rm los}=c\frac{0.6}{689.1}\simeq261.0\,\mathrm{km\,s^{-1}},\qquad
\boxed{v_{\max}=\frac{v_{\rm los}}{\sin i}\simeq269.6\,\mathrm{km\,s^{-1}}.}
$$
The circular-orbit and thin-disk assumptions neglect intrinsic thickness, warps and noncircular gas motion.
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