Here is a proof using a Fitch-style proof checker.
The first two lines contain the premises.
Since the goal is a conditional, I assumed the antecedent, S, in a subproof starting on line 3. My goal was to reach the consequent, Q v R, which I did on line 13. This allowed me to discharge the assumption and close the subproof by introducing a conditional to complete the proof.
To get to line 13, I assumed the negation of that consequent I wanted to prove by starting a new subproof. I aimed to reach a contradiction which I did in line 12. Using the disjunctive syllogism (DS) on line 8 allowed me to cut short testing the two cases of the disjunction on line 6.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/