The given equation of the ellipse is:

$\frac{(x - 2)^2}{6} + \frac{(y + 3)^2}{8} = 1$

Step 1: Multiply both sides of the equation by the least common denominator (LCD).

The denominators are 6 and 8, so the least common denominator is 24. Multiply through by 24 to eliminate the fractions:

$24 \left(\frac{(x - 2)^2}{6} + \frac{(y + 3)^2}{8}\right) = 24(1)$

Simplify the equation:

$4(x - 2)^2 + 3(y + 3)^2 = 24$

Step 2: Expand the squared terms.

Expand $(x - 2)^2$ and $(y + 3)^2$:

$4((x - 2)(x - 2)) + 3((y + 3)(y + 3)) = 24$

$4(x^2 - 4x + 4) + 3(y^2 + 6y + 9) = 24$

Step 3: Distribute the constants.

Distribute 4 to the terms in the first parenthesis and 3 to the terms in the second parenthesis:

$4x^2 - 16x + 16 + 3y^2 + 18y + 27 = 24$

Step 4: Simplify the equation.

Combine like terms:

$4x^2 + 3y^2 - 16x + 18y + 43 = 24$

Step 5: Move the constant to the other side.

Subtract 24 from both sides:

$4x^2 + 3y^2 - 16x + 18y + 19 = 0$

This is the general form of the ellipse equation.

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Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:

1. How do you convert the general form back to standard form?
2. What are the properties of an ellipse from the general form equation?
3. How would you find the center of this ellipse?
4. Can you determine the lengths of the major and minor axes from this equation?
5. What is the method to graph this ellipse using the general form equation?

Tip: Always verify if the constant is positive before dividing the entire equation when simplifying quadratic equations.