Determine the polynomial P(X) satisfying simoultaneously the conditions:
a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal
to the fourth power of quotient.
b) P(−2) + P(2) = −34.
1. **Understanding the first condition:**
   - We are given that the remainder obtained when dividing \( P(x) \) by \( x^3 - 2 \) is equal to the fourth power of the quotient.
   - Let \( Q(x) \) be the quotient when \( P(x) \) is divided by \( x^3 - 2 \). Then we can write:
     \[
     P(x) = (x^3 - 2)Q(x) + R(x)
     \]
   - According to the problem, the remainder \( R(x) \) is equal to \( Q(x)^4 \). Therefore:
     \[
     P(x) = (x^3 - 2)Q(x) + Q(x)^4
     \]

2. **Degree considerations:**
   - The degree of \( x^3 - 2 \) is 3. Therefore, the degree of \( (x^3 - 2)Q(x) \) is \( \deg(Q(x)) + 3 \).
   - Since \( R(x) = Q(x)^4 \), the degree of \( R(x) \) is \( 4 \cdot \deg(Q(x)) \).
   - For the polynomial \( P(x) \) to be consistent, the degrees of both parts must match. This implies that \( \deg(Q(x)) \) must be zero (i.e., \( Q(x) \) is a constant).

3. **Simplifying with \( Q(x) = a \):**
   - Let \( Q(x) = a \), where \( a \) is a constant. Then:
     \[
     P(x) = (x^3 - 2)a + a^4
     \]
   - Simplifying, we get:
     \[
     P(x) = a x^3 - 2a + a^4
     \]

4. **Using the second condition:**
   - We are given \( P(-2) + P(2) = -34 \).
   - Calculate \( P(2) \):
     \[
     P(2) = a(2^3) - 2a + a^4 = 8a - 2a + a^4 = 6a + a^4
     \]
   - Calculate \( P(-2) \):
     \[
     P(-2) = a(-2^3) - 2a + a^4 = -8a - 2a + a^4 = -10a + a^4
     \]
   - Adding these, we get:
     \[
     P(2) + P(-2) = (6a + a^4) + (-10a + a^4) = -4a + 2a^4
     \]
   - According to the problem:
     \[
     -4a + 2a^4 = -34
     \]
   - Simplifying, we get:
     \[
     2a^4 - 4a = -34
     \]
     \[
     a^4 - 2a = -17
     \]
     \[
     a^4 - 2a + 17 = 0
     \]

5. **Solving the polynomial equation:**
   - We need to solve \( a^4 - 2a + 17 = 0 \).
   - By inspection or using numerical methods, we can determine that this polynomial has no real roots. This can be confirmed by analyzing the function \( f(a) = a^4 - 2a + 17 \) and noting that it does not cross the x-axis.

Therefore, there is no real value of \( a \) that satisfies the given conditions.

The final answer is \(\boxed{\text{No such polynomial}}\)