Let's simplify the expression ((a+1)^2 + (a-1)^2) / (3a^2 + 3) step by step.

Expanding the numerators:
((a+1)^2 + (a-1)^2) = (a^2 + 2a + 1) + (a^2 - 2a + 1) = 2a^2 + 2

Expanding the denominator:
3a^2 + 3

Now we can rewrite the expression as:
(2a^2 + 2) / (3a^2 + 3)

Factor out a 2 from the numerator:
2(a^2 + 1)

Factor out a 3 from the denominator:
3(a^2 + 1)

Now our expression simplifies to:
2(a^2 + 1) / 3(a^2 + 1)

Finally, we can cancel out the common factor (a^2 + 1) from the numerator and the denominator:
2/3

Therefore, ((a+1)^2 + (a-1)^2) / (3a^2 + 3) simplifies to 2/3.