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Step-by-Step Solution
Step 1: Express the polynomial in standard form
Let the required polynomial be
$P(x) = ax^2 + bx + c.$
Step 2: Apply the condition P(0) = 1
Substitute $x = 0$ in the polynomial:
$$
P(0) = a \cdot 0^2 + b \cdot 0 + c = c.
$$
Given $P(0) = 1$, we get
$$
c = 1.
$$
Thus,
$$
P(x) = ax^2 + bx + 1.
$$
Step 3: Use the Remainder Theorem for x − 1
When $P(x)$ is divided by $(x - 1)$, the remainder is $P(1)$. Given that this remainder is 4, we have:
$$
P(1) = a(1)^2 + b(1) + 1 = a + b + 1 = 4.
$$
This simplifies to:
$$
a + b = 3.
\quad\quad (1)
$$
Step 4: Use the Remainder Theorem for x + 1
Similarly, when $P(x)$ is divided by $(x + 1)$, the remainder is $P(-1)$. Given that this remainder is 6, we have:
$$
P(-1) = a(-1)^2 + b(-1) + 1 = a - b + 1 = 6.
$$
This simplifies to:
$$
a - b = 5.
\quad\quad (2)
$$
Step 5: Solve for a and b
From equations (1) and (2):
$$
\begin{cases}
a + b = 3, \\
a - b = 5.
\end{cases}
$$
Adding these two equations, we get:
$$
2a = 8 \quad \Rightarrow \quad a = 4.
$$
Substituting $a = 4$ into $a + b = 3$ gives:
$$
4 + b = 3 \quad \Rightarrow \quad b = -1.
$$
Therefore,
$$
P(x) = 4x^2 - x + 1.
$$
Step 6: Evaluate P(-2)
To find $P(-2)$:
$$
P(-2) = 4(-2)^2 - (-2) + 1 = 4 \cdot 4 + 2 + 1 = 16 + 2 + 1 = 19.
$$
Hence,
$$
P(-2) = 19.
$$
Conclusion
The correct answer is $\displaystyle P(-2) = 19.$
