The basic conversion to a second-order linear equation is straightforward.
The linear equation is holonomic, but not easy to solve apparently. DSolve returns a DifferentialRoot[] solution.
Block[{q, r, s},
{q[0], q[1], q[2]} =
CoefficientList[
y'[x] /. First@Solve[2 x*y'[x] == (x + 2) y[x]^2 - (x + 4), y'[x]],
y[x]];
s = q[0] q[2];
r = q[1] + D[q[2], x]/q[2];
usol = DSolve[u''[x] - r*u'[x] + s*u[x] == 0, u, x];
ysol = {y[x] -> -u'[x]/u[x]/q[2]} /. First@usol
]
If you want a derivation of the Kummer function expression, you might have better luck asking on a mathematics site.
