OG 2017 Online Two Part Analysis Problem

Here, we can translate the sequence notation
as the following:
For any given nth term (Sn), the value of the term two places above it (S(n+2)) will be equal to the nth term squared minus the value of the term one above the nth term (S(n-1)).

Or in other words...


Since S4 depends on knowing the values of both S2 and S3, and S3 depends on knowing the values of S1 and S2, let's begin by testing the possible given values of S1:



Now, generate the value of S3 for each by squaring S1 then subtracting S2:


Now, generate S4 for each by squaring S2 then subtracting S3:


Now we need to find a value for S1 that would have a matching value for S4.


Since -142, -47, and 1 are not options for S4, eliminate -12, -7, and -1 as options for S1. S4 = 0 is an option... but we're told to select 2 different values, so this must not be the answer.
The only valid match is:
S1 = -3
S4 = -7