This question was on my extra credit test, and I'm having problems with parts of it:

Graph the rational function below:

Solve ALGEBRAICALLY X and Y intercepts, all asymptotes, and local extrema (min and max) points and any intersection point with slant asymptote. Find domain, range, intervals where the function is <0 and >0, increasing and decreasing using interval notation.

I understand how to do most of this but not how to find the local extrema algebraically. I've tried completing the square and I can't quite get it to work.

So far I've gotten that the y int is 1.5, there's no x int, the slant asymptote is y = x - 4, the vertical asymptote is 2, and there's no horizontal asymptote. I'm not sure if all that information is correct but it's what I've gotten so far, and now I'm sort of stuck on the max and min points.

Thank you for any help with any of the aspects of this problem!