Question

(2.60 x 10^3 x 0.897) + 2.69 (with correct # of sig figs)

IS the answer 2334.89

Answer 1

Right digits, yes. Wrong answer.
You have 3 s.f. in 2.60E3 and 3 s.f. in 0.897 and 2.69.
Multiply 2.60E3*0.897 = 2332.2. If you add 2.69 the sum is the value you obtained of 2334.89. But you may have only 3 places which means you have 2330. The way you write that, since the zero is not significant, is 2.33E3.
I think the purist would do it this way.
2.60E3*0.869 = 2332 which becomes
2.33E3. Then add
0.00269E3 which is insignificant compared to the 2.33E3 so the answer remains 2.33E3.