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?

Let p(x) = ax^2 + bx + c. From p(0) = 1, we know c = 1. By the polynomial remainder theorem, p(1) = a + b + 1 = 4 and p(-1) = a - b + 1 = 6. Solving these two equations gives a = 4 and b = -1, making the polynomial p(x) = 4x^2 - x + 1. Evaluating p(-2) yields 4(-2)^2 - (-2) + 1 = 16 + 2 + 1 = 19.

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