First look at the ones digit. Something that ends in 7 plus something that ends in 6 usually ends in 3. But in this case the answer ends in 5, so we must not be working in base 10! Instead, 6+7=13 must be 5 plus a multiple of our base. So either it's base 8, base 4, or base 2. But it can't be base 4 or base 2, since some of the digits are bigger than 4. The only possibility left is base 8.
So is it really base 8? Let's check:
6+7 = 13 (base 10) which is 15 (base 8). So write down a 5 and carry the one.
3+7+1 = 11 (base 10), which is 13 (base 8). Write down a 3 and carry the one.
1+2+1=4.
So 137 (base 8) + 276 (base 8) = 435 (base 8). So far so good!
Now do the other addition base 8:
1+2=3, so the last digit is 3
3+7=10 (base 10) = 12 (base 8), so the next digit is 2. Carry the 1.
7+6+1 = 14 (base 10) = 16 (base 8).