Electronic – I set up an amplifier (LF351) for a gain of 5.7, but it is saturating. Do I have to utilize the offset null pins

I'll give you a hint here. You can calculate the Thevenin equivalent of the what is seen looking into R2.

By inspection its source resistance is:

\$R_T = R_2+(R_3||R_4)\$, and the open-circuit source voltage is:

\$V_{OUT}\cdot \frac {R_3}{R_3+R_4} \$

So the gain from the (virtual) Thevenin source is just -Rt/R1, but the output voltage must be higher by a factor of \$\frac {R_3+R_4}{R_3} \$.

You can then calculate the gain and the output voltage. Chances are if this is actually built the op-amp will saturate at the negative rail.

There are other, more general, ways to solve a problem like this (write the equations out) but with this way the answer drops out pretty easily.

Yes the circuit would work without either resistor but there may be benefits from having them. The bias currents into the rusty old 741's inputs is about 100 nA and this will produce an error on top of the 5V produced by R1 and R2 but only about 0.25 mV so trying to balance that current with R5 isn't really a big deal.

However, R5 does somewhat protect the 741's input from unknown loading effects on the output. You say that this circuit is used in the minimoog but if that 5V were also available as an output then R5 could offer the 741 some protection from external problems.

Ditto R3 - if someone shorted the -5V rail then R3 would limit the current taken from the 741 into the base and give it a level of protection. R3 also limits the current that can be delivered at the emitter of Q1 too.

The main problem here is that this circuit doesn't have any context and, that back in the late 60s/early 70s, who knows what went through the mind of the designer. Op-amps were beginning to be used commercially but their reputation wasn't as great as it is now and maybe the designer thought more about protecting the device against unreasonable "what-if" scenarios that we would today.