Statement 1: a - b = 4
a = 4 + b, or b = a - 4
$$By\ substituting\ a^2+b^2=\left(4+b\right)^2+b^2$$
$$=\left(4+b\right)\left(4+b\right)+b^2$$
$$=2\left(b^2+4+18\right)\ $$
OR
$$a^2+b^2=a^2+\left(a-4\right)^2$$
$$=a^2+\left(a-4\right)\left(a-4\right)$$
$$=2\left(a^2-4a+8\right)$$
The exact value of 'a' and 'b' is unknown; hence, the target question cannot be answered. Therefore, statement 1 is NOT SUFFICIENT.
$$Statement\ 2:\ a^2b^3\ is\ divisible\ by\ 14$$
For the given expression to be divisible by 14. It is either 'a' has 7 as a prime factor or 'b' has 7 as a prime factor. With these multiple alternatives, there is no definite answer. Hence, statement 2 is NOT SUFFICIENT.
Combining both statements together
From statement 1: a - b = 4
From statement 2: a has to be divisible by 7 since at least one of them will have 7 as part of its prime factors.
Note that both 'a' and 'b' cannot be 7 as a-b will not = 4.
If a=7, then b=3
$$So,\ \frac{a^2+b^2}{7}=\frac{7^2+3^2}{7};\ remainder\ is\ 2$$
If a=1, then b=7
$$So,\ \frac{a^2+b^2}{7}=\frac{1^2+7^2}{7};\ remainder\ is\ also\ 2$$
$$Therefore,\ when\ a^2+b^2\ is\ divided\ by\ 7,\ the\ remainder\ is\ 2$$
Hence, both statements combined are SUFFICIENT; ANSWER = C
