Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?

$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $
1. Let \( P(x) = ax^3 + bx^2 + cx + d \). We are given that \( P(-1) = -9 \).
2. Substituting \( x = -1 \) into \( P(x) \), we get:
   \[
   P(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d
   \]
3. We need \( -a + b - c + d = -9 \). Rearranging, we get:
   \[
   b + d = a + c + 9
   \]
4. Let \( s = a + c + 9 \). Then \( b + d = s \). The possible values of \( s \) range from 9 to 18 (since \( a, b, c, d \) are digits from 0 to 9).
5. For each fixed \( s \), we need to count the number of valid pairs \((a, c)\) and \((b, d)\) such that:
   \[
   a + c = s - 9 \quad \text{and} \quad b + d = s
   \]
6. The number of pairs \((a, c)\) such that \( a + c = s - 9 \) is \( s - 8 \) (since \( a \) and \( c \) are digits from 0 to 9).
7. The number of pairs \((b, d)\) such that \( b + d = s \) is \( 19 - s \) (since \( b \) and \( d \) are digits from 0 to 9).
8. Therefore, for each \( s \), the number of valid polynomials is:
   \[
   (s - 8)(19 - s)
   \]
9. We need to sum this product over all possible values of \( s \) from 9 to 18:
   \[
   \sum_{s=9}^{18} (s - 8)(19 - s)
   \]
10. Simplifying the sum:
    \[
    \sum_{s=9}^{18} (s - 8)(19 - s) = \sum_{s=0}^{9} (s + 1)(10 - s)
    \]
11. Calculating each term:
    \[
    \begin{aligned}
    & (0 + 1)(10 - 0) = 1 \cdot 10 = 10 \\
    & (1 + 1)(10 - 1) = 2 \cdot 9 = 18 \\
    & (2 + 1)(10 - 2) = 3 \cdot 8 = 24 \\
    & (3 + 1)(10 - 3) = 4 \cdot 7 = 28 \\
    & (4 + 1)(10 - 4) = 5 \cdot 6 = 30 \\
    & (5 + 1)(10 - 5) = 6 \cdot 5 = 30 \\
    & (6 + 1)(10 - 6) = 7 \cdot 4 = 28 \\
    & (7 + 1)(10 - 7) = 8 \cdot 3 = 24 \\
    & (8 + 1)(10 - 8) = 9 \cdot 2 = 18 \\
    & (9 + 1)(10 - 9) = 10 \cdot 1 = 10 \\
    \end{aligned}
    \]
12. Summing these values:
    \[
    10 + 18 + 24 + 28 + 30 + 30 + 28 + 24 + 18 + 10 = 220
    \]

The final answer is \(\boxed{220}\)