Help needed with shear and moment problem

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Please refer to the link above. This assignment will be due before I get any response, but I would still like to understand the problem thoroughly. Understanding is more valuable to me than a grade.

I'm having a little bit of trouble figuring this one out.

First of all, I made four cuts: one before the downward force of 500N, one after that force but before the 750N/m (the professor labeled q1 and q2 with the wrong units) distributed load, one in between the 750N/m distributed and one in between the linearly distributed load. Are these the correct cuts to make?

Assuming those are the correct cuts (and please let me know if they aren't) for the first section my equations are V(x)=5820+500x-(25/3)x^2 and M(x)=5820x+250x^2-(25/9)x^3. Is this correct?

And finally (and most importantly since this is the most confusing to me) how do I handle the problem for the second portion (30m<x<60m)? Do I leave the distributed load as it is? Do I replace the left portion of the distributed load (0<x<30m) with the resultant force and treat the right portion as a new distributed load?

I know this on is a doozy, but your help will be much appreciated.