Question 1209770
Let's solve this equation step-by-step:

**1. Simplify the Equation:**

Let a = x - 4. Then:

* x - 3 = a + 1
* x - 5 = a - 1

Substitute these into the equation:

(a + 1)^4 + (a - 1)^4 = -8 + 6(a + 1)(a - 1)^3 - 11(a + 1)^3 (a - 1)

Expand the terms:

* (a + 1)^4 = a^4 + 4a^3 + 6a^2 + 4a + 1
* (a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1
* (a + 1)^4 + (a - 1)^4 = 2a^4 + 12a^2 + 2

* (a - 1)^3 = a^3 - 3a^2 + 3a - 1
* (a + 1)^3 = a^3 + 3a^2 + 3a + 1

Substitute these into the right side:

-8 + 6(a + 1)(a^3 - 3a^2 + 3a - 1) - 11(a^3 + 3a^2 + 3a + 1)(a - 1)
-8 + 6(a^4 - 3a^3 + 3a^2 - a + a^3 - 3a^2 + 3a - 1) - 11(a^4 + 3a^3 + 3a^2 + a - a^3 - 3a^2 - 3a - 1)
-8 + 6(a^4 - 2a^3 + 2a - 1) - 11(a^4 + 2a^3 - 2a - 1)
-8 + 6a^4 - 12a^3 + 12a - 6 - 11a^4 - 22a^3 + 22a + 11
-5a^4 - 34a^3 + 34a - 3

Now, the equation becomes:

2a^4 + 12a^2 + 2 = -5a^4 - 34a^3 + 34a - 3
7a^4 + 34a^3 + 12a^2 - 34a + 5 = 0

**2. Factor the Equation:**

Notice that the coefficients are symmetric. This suggests that we can factor by grouping.

Let's divide by a^2:

7a^2 + 34a + 12 - 34/a + 5/a^2 = 0
7(a^2 + 5/7a^2) + 34(a - 1/a) + 12 = 0

Let b = a - 1/a. Then b^2 = a^2 - 2 + 1/a^2, so a^2 + 1/a^2 = b^2 + 2

7(b^2 + 2) + 34b + 12 = 0
7b^2 + 14 + 34b + 12 = 0
7b^2 + 34b + 26 = 0

Using the quadratic formula, we have:

b = (-34 ± √(34^2 - 4*7*26)) / (2*7)
b = (-34 ± √(1156 - 728)) / 14
b = (-34 ± √428) / 14
b = (-34 ± 2√107) / 14
b = (-17 ± √107) / 7

**3. Solve for a:**

* a - 1/a = b
* a^2 - 1 = ba
* a^2 - ba - 1 = 0

Using the quadratic formula again:

* a = (b ± √(b^2 + 4)) / 2

**4. Solve for x:**

* x = a + 4

**5. Numerical Solutions:**

Using numerical methods (like those used by the Python code), we find the real roots:

* x ≈ 2.445
* x ≈ 4.690

**Final Answer:**

The real solutions are approximately x ≈ 2.445 and x ≈ 4.690.