In utility theory, the aim of the individual is to maximise the expected utility. As you have shown in the last line, the expected utility for an individual with a quadratic utility function is given as:
So we should try to maximise this above function. But as you can see, this function is only dependent on the first two moments of the distribution of R (i.e. E(R) and E(R^2)) . So they do not care about the higher moments of the distribution.
In Mean Portfolio Theory, the investor's choice is simply determined by the mean and the variance. You should recognise that the variance can be calculated using these first two moments. Hence. the choice the investor makes will be the same one as the individual who has a quadratic utility function.
Hope this helps!