Find with all integers $n$ when  $|n^3 - 4n^2 + 3n - 35|$  and $|n^2 + 4n + 8|$  are prime numbers.
To find all integers \( n \) such that both \( |n^3 - 4n^2 + 3n - 35| \) and \( |n^2 + 4n + 8| \) are prime numbers, we will analyze each expression separately and then check for common solutions.

1. **Analyze \( |n^3 - 4n^2 + 3n - 35| \):**
   - We need \( n^3 - 4n^2 + 3n - 35 \) to be a prime number or its negative to be a prime number.
   - Let's denote \( P(n) = n^3 - 4n^2 + 3n - 35 \).

2. **Analyze \( |n^2 + 4n + 8| \):**
   - We need \( n^2 + 4n + 8 \) to be a prime number or its negative to be a prime number.
   - Let's denote \( Q(n) = n^2 + 4n + 8 \).

3. **Check for small integer values of \( n \):**
   - For \( n = 0 \):
     \[
     P(0) = 0^3 - 4 \cdot 0^2 + 3 \cdot 0 - 35 = -35 \quad (\text{not prime})
     \]
     \[
     Q(0) = 0^2 + 4 \cdot 0 + 8 = 8 \quad (\text{not prime})
     \]
   - For \( n = 1 \):
     \[
     P(1) = 1^3 - 4 \cdot 1^2 + 3 \cdot 1 - 35 = 1 - 4 + 3 - 35 = -35 \quad (\text{not prime})
     \]
     \[
     Q(1) = 1^2 + 4 \cdot 1 + 8 = 1 + 4 + 8 = 13 \quad (\text{prime})
     \]
   - For \( n = 2 \):
     \[
     P(2) = 2^3 - 4 \cdot 2^2 + 3 \cdot 2 - 35 = 8 - 16 + 6 - 35 = -37 \quad (\text{prime})
     \]
     \[
     Q(2) = 2^2 + 4 \cdot 2 + 8 = 4 + 8 + 8 = 20 \quad (\text{not prime})
     \]
   - For \( n = 3 \):
     \[
     P(3) = 3^3 - 4 \cdot 3^2 + 3 \cdot 3 - 35 = 27 - 36 + 9 - 35 = -35 \quad (\text{not prime})
     \]
     \[
     Q(3) = 3^2 + 4 \cdot 3 + 8 = 9 + 12 + 8 = 29 \quad (\text{prime})
     \]
   - For \( n = 4 \):
     \[
     P(4) = 4^3 - 4 \cdot 4^2 + 3 \cdot 4 - 35 = 64 - 64 + 12 - 35 = -23 \quad (\text{prime})
     \]
     \[
     Q(4) = 4^2 + 4 \cdot 4 + 8 = 16 + 16 + 8 = 40 \quad (\text{not prime})
     \]
   - For \( n = 5 \):
     \[
     P(5) = 5^3 - 4 \cdot 5^2 + 3 \cdot 5 - 35 = 125 - 100 + 15 - 35 = 5 \quad (\text{prime})
     \]
     \[
     Q(5) = 5^2 + 4 \cdot 5 + 8 = 25 + 20 + 8 = 53 \quad (\text{prime})
     \]

4. **Conclusion:**
   - The only integer \( n \) for which both \( |n^3 - 4n^2 + 3n - 35| \) and \( |n^2 + 4n + 8| \) are prime numbers is \( n = 5 \).

The final answer is \( \boxed{n = 5} \).