To identify the type of conic section each equation represents, we need to analyze the general forms and coefficients of the $x^2$ and $y^2$ terms. Let's break it down for each equation:

1. $x^2 = -2y^2 + 8$
   This equation can be rewritten as $x^2 + 2y^2 = 8$. Since the coefficients of $x^2$ and $y^2$ have opposite signs, this is a hyperbola.

2. $2x^2 + 4y^2 - 3x = 14$
   This equation has both $x^2$ and $y^2$ with the same signs, indicating that it is an ellipse (if both are positive). The linear term $-3x$ indicates a translation of the conic.

3. $4x^2 + x - 18y^2 + 12y = 5$
   Here, $x^2$ and $y^2$ have opposite signs, so it’s a hyperbola.

4. $-x^2 + 8x - y = -4y^2$
   This can be rewritten as $-x^2 + 4y^2 + 8x = y$. Again, opposite signs for $x^2$ and $y^2$ indicate a hyperbola.

5. $3x^2 - 15y^2 + 2x - 3y = 4$
   Opposite signs for $x^2$ and $y^2$, so this is a hyperbola.

6. $5x^2 + 5y - 8 = -5y^2 - 2x$
   This simplifies into an equation where $x^2$ and $y^2$ terms have opposite signs. Thus, it's a hyperbola.

7. $-x^2 + 5y^2 + 3y = 5$
   This has opposite signs for $x^2$ and $y^2$, confirming that it is a hyperbola.

Would you like further details on each conic, or need assistance with another part of the problem? Here are some related questions:

1. How do you classify conic sections based on their general equation?
2. What are the key characteristics of a hyperbola?
3. How does translating a conic section affect its equation?
4. What is the difference between an ellipse and a circle in terms of their equation?
5. Can a linear term in a conic section equation affect the type of conic?

Tip: Always look at the signs and coefficients of the squared terms ($x^2$ and $y^2$) to classify conic sections.