Let p(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x - 1 and it leaves remainder 6 when divided by x + 1; then which one is correct?

p(x) = ax² + bx + c
p(0) = 1 = c = 1
p(1) = 4
p(-1) = 6
[a + b + c = 4
a - b + c = 6]
Subtracting the two equations:
2b = -2
b = -1
Substituting b = -1 in a + b + c = 4:
a - 1 + 1 = 4
a = 4
Therefore, p(x) = 4x² - x + 1
p(2) = 4(2)² - 2 + 1 = 16 - 2 + 1 = 15
However, none of the options match this result. Let's re-examine the equations:
Given p(1) = 4 and p(-1) = 6
Let p(x) = ax² + bx + c
p(0) = 1 => c = 1
p(1) = a + b + 1 = 4 => a + b = 3
p(-1) = a - b + 1 = 6 => a - b = 5
Adding the two equations:
2a = 8
a = 4
Substituting a = 4 into a + b = 3:
4 + b = 3
b = -1
So p(x) = 4x² - x + 1
p(2) = 4(2)² - 2 + 1 = 16 - 2 + 1 = 15
There must be a mistake in the given options. None of the options are correct based on the given information.