Let $$S=\left\{x^{3}+ax^{2}+bx+c:a,b,c, \in N \text{ and }a,b,c \leq 20\right\}$$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $$x^{2}+2$$, is

We need to find the number of polynomials $$x^3 + ax^2 + bx + c$$ (with $$a,b,c \in \{1,...,20\}$$) that are divisible by $$x^2 + 2$$.

If $$x^2 + 2$$ divides $$x^3 + ax^2 + bx + c$$, then:

$$x^3 + ax^2 + bx + c = (x^2 + 2)(x + a) = x^3 + ax^2 + 2x + 2a$$

Comparing coefficients:

$$b = 2$$ and $$c = 2a$$

$$b = 2$$ (fixed, and $$2 \leq 20$$ ✓)

$$c = 2a$$, with $$c \leq 20$$ and $$a \leq 20$$

$$2a \leq 20 \implies a \leq 10$$

Also $$a \geq 1$$

So $$a$$ can be 1, 2, 3, ..., 10 → 10 values.

Therefore, the number of polynomials is Option 1: 10.