Option 3 : -6

__Concept:__

*Remainder Theorem: *Let p(x) be any polynomial of degree greater than or equal to one and 'a' be any real number. If p(x) is divided by (x - a), then the remainder is equal to p(a).

**Calculation:**

When p(x) is divided by (x - 1) and (x + 2), the remainders are 4 and 22 respectively.

∴ p(1) = 4 and p(-2) = 22

So, we have, p(1) = 4

⇒ 2(1)^{4} + (1)3 + 3(1)2 - s(1) + t = 4

⇒ 2 + 1 + 3 - s + t = 4

⇒ 6 - s + t = 4

⇒ -s + t = -2 ----(1)

Similarly, p(-2) = 22

⇒ 2(-2)4 + (-2)3 + 3(-2)2 - s(-2) + t = 22

⇒ 32 - 8 + 12 + 2s + t = 22

⇒ 36 + 2s + t = 22

⇒ 2s + t = -14 ----(2)

On solving equations (1) & (2), we get,

s = -4 and t = -6

On putting the values of 's' and 't' in p(x), we get,

p(x) = 2x4 + x3 + 3x2 + 4x - 6

Now, if we divide this polynomial p(x) by (x + 1), the remainder would be p(-1).

So, Remainder = p(-1) = 2(-1)4 + (-1)3 + 3(-1)2 + 4(-1) - 6

⇒ 2 - 1 + 3 - 4 - 6

⇒ -6

**Hence, the remainder when p(x) is divided by (x + 1) is -6.**