Identify the polynomial and the divisor.

Another day — let's solve it.

Step 1: Identify the polynomial and the divisor. The polynomial is $f(x) = x^4 - 2x^3 - 6x^2 - 2x + 1$. The divisor is $x+2$.

Step 2: Apply the Remainder Theorem. The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x-a$, the remainder is $f(a)$. In this case, the divisor is $x+2$, which can be written as $x - (-2)$. So, $a = -2$. Therefore, the remainder is $f(-2)$.

Step 3: Substitute $x = -2$ into the polynomial $f(x)$. $f(-2) = (-2)^4 - 2(-2)^3 - 6(-2)^2 - 2(-2) + 1$

Step 4: Calculate each term. $(-2)^4 = 16$ $-2(-2)^3 = -2(-8) = 16$ $-6(-2)^2 = -6(4) = -24$ $-2(-2) = 4$ $+1$

Step 5: Sum the calculated terms to find the remainder. $f(-2) = 16 + 16 - 24 + 4 + 1$ $f(-2) = 32 - 24 + 4 + 1$ $f(-2) = 8 + 4 + 1$ $f(-2) = 13$

The remainder is $13$.

The final answer is $\boxed{13}$.

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