Here is the question:
((p->q) and (r->s) and (p or r)) -> (q or s)
How would you prove that this is tautology? Using natural deduction?
Since one wants to prove that this is a tautology one would use a truth table, that is, one would use a semantic approach to solving the problem in truth-functional logic. The semantics refers to the true or false valuations of the atomic sentences.
Since there are four atomic sentences, p, q, r and s, the truth table should have 16 lines as Araucaria's answer showed. Here is a proof using Stanford's truth table tool:
That should be the end of it, however, one could approach the problem from a syntactic, rather than a semantic, approach. Then one would use something like natural deduction. One would derive the result using inference rules. Considering that this is described as a "tautology", that may not be what the assignment is asking for.
Here is one way to do that using Klement's Fitch-style natural deduction proof checker:
I used the following rules which can be found described in forall x: conjunction elimination (∧E), conditional elimination (→E), disjunctive introduction (DS), law of excluded middle (LEM) and (conditional introduction (→I).
Using this syntactic approach for a tautology is questionable unless one knows that the two approaches, truth tables and derivations give the same results.
This goes beyond the original question, but it is covered with an outline of a proof in forall x in Chapter 20. It is covered in more detail in Lemmon's Beginning Logic, pages 75-91.
Araucaria's answer: https://philosophy.stackexchange.com/a/33727/29944
Lemmon, E. J. (1971). Beginning logic. CRC Press.
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/ Wikipedia, "Fitch notation" https://en.wikipedia.org/wiki/Fitch_notation
Stanford's Truth Table Tool: http://web.stanford.edu/class/cs103/tools/truth-table-tool/