Question 1209759
Let's analyze the given equation step-by-step:

**1. Simplify the Trigonometric Expression:**

* **Numerator:**
    * 3cos(2x) + cos³(2x) = cos(2x)(3 + cos²(2x))

* **Denominator:**
    * cos⁶(x) - sin⁶(x) = (cos²(x))³ - (sin²(x))³
    * Using a³ - b³ = (a - b)(a² + ab + b²):
        * (cos²(x) - sin²(x))(cos⁴(x) + cos²(x)sin²(x) + sin⁴(x))
    * cos²(x) - sin²(x) = cos(2x)
    * cos⁴(x) + sin⁴(x) = (cos²(x) + sin²(x))² - 2cos²(x)sin²(x) = 1 - 2cos²(x)sin²(x)
    * cos⁶(x) - sin⁶(x) = cos(2x)(1 - 2cos²(x)sin²(x) + cos²(x)sin²(x))
    * cos⁶(x) - sin⁶(x) = cos(2x)(1 - cos²(x)sin²(x))
    * cos²(x)sin²(x) = (1/4)(2sin(x)cos(x))² = (1/4)sin²(2x)
    * cos⁶(x) - sin⁶(x) = cos(2x)(1 - (1/4)sin²(2x))

* **Substitute into the Fraction:**
    * [cos(2x)(3 + cos²(2x))] / [cos(2x)(1 - (1/4)sin²(2x))]
    * (3 + cos²(2x)) / (1 - (1/4)sin²(2x))

* **Use cos²(2x) = 1 - sin²(2x):**
    * (3 + 1 - sin²(2x)) / (1 - (1/4)sin²(2x))
    * (4 - sin²(2x)) / (1 - (1/4)sin²(2x))
    * Let y = sin²(2x):
        * (4 - y) / (1 - y/4) = (4 - y) / ((4 - y)/4) = 4

**2. Simplify the Equation:**

* The equation becomes: 4 = x³ - x² + 6
* x³ - x² + 2 = 0

**3. Find the Roots:**

* Let's try x = -1: (-1)³ - (-1)² + 2 = -1 - 1 + 2 = 0
* Therefore, x = -1 is a root.
* We can perform polynomial division to find the other factors:
    * (x³ - x² + 2) / (x + 1) = x² - 2x + 2
* The quadratic x² - 2x + 2 has discriminant (-2)² - 4(1)(2) = 4 - 8 = -4, which is negative. Therefore, it has no real roots.

**4. Find the Sum of Real Solutions:**

* The only real solution is x = -1.

**Final Answer:**

The sum of real solutions is -1.