Question 1209661
Here's how to identify the conic section:

1. **Rearrange the equation:**  Move all terms to one side:

   3x² + y² + 9x - 5y - 20 - 8x² - 6x - 47 = 0
   -5x² + 3x + y² - 5y - 67 = 0

2. **Multiply by -1 (optional, but makes some calculations easier):**

   5x² - 3x - y² + 5y + 67 = 0

3. **Complete the square for both x and y:**

   5(x² - (3/5)x) - (y² - 5y) + 67 = 0
   5(x² - (3/5)x + (3/10)²) - 5(3/10)² - (y² - 5y + (5/2)²) + (5/2)² + 67 = 0
   5(x - 3/10)² - 5(9/100) - (y - 5/2)² + 25/4 + 67 = 0
   5(x - 3/10)² - 9/20 - (y - 5/2)² + 25/4 + 67 = 0
   5(x - 3/10)² - (y - 5/2)² + 67 - 9/20 + 125/20 = 0
   5(x - 3/10)² - (y - 5/2)² + 67 + 116/20 = 0
   5(x - 3/10)² - (y - 5/2)² + 67 + 5.8 = 0
   5(x - 3/10)² - (y - 5/2)² + 72.8 = 0
   5(x - 3/10)² - (y - 5/2)² = -72.8

4. **Divide by -72.8:**

   [5(x - 3/10)²] / -72.8 - [(y - 5/2)²] / -72.8 = 1
   [(x - 3/10)²] / (-72.8/5) - [(y - 5/2)²] / -72.8 = 1
   [(x - 3/10)²] / -14.56 - [(y - 5/2)²] / -72.8 = 1

5. **Analyze the equation:**

   The equation is in the form:

   (x²/a²) - (y²/b²) = 1  (Hyperbola)
   OR
   -(x²/a²) + (y²/b²) = 1 (Hyperbola)

Since both the x² and y² terms are present and have *opposite* signs, the equation represents a **hyperbola**.  The negative signs simply determine the orientation of the hyperbola (whether it opens horizontally or vertically). Since the x^2 term is negative, this is a hyperbola that opens up and down.