To solve for $n$ in this problem, we need to use the fact that the sum of the interior angles of a four-sided polygon (quadrilateral) is always 360°.

The four angles are given as:

• $\angle A = 4n$
• $\angle B = 9n + 2$
• $\angle C = 8n - 12$
• $\angle D = 5n + 6$

The sum of the angles is: $4n + (9n + 2) + (8n - 12) + (5n + 6) = 360°$

Now, let's combine like terms:

$4n + 9n + 8n + 5n = 26n$ $2 - 12 + 6 = -4$

So, the equation becomes: $26n - 4 = 360$

Now, solve for $n$:

1. Add 4 to both sides: $26n = 364$
2. Divide both sides by 26: $n = \frac{364}{26} = 14$

So, $n = 14$.

Do you want more details or have any questions? Here are some related questions to explore further:

1. What would happen if the sum of angles in a polygon were different than 360°?
2. How do you derive the formula for the sum of interior angles of a polygon?
3. How can you calculate the interior angles of a regular polygon?
4. What happens if the polygon has more than four sides?
5. How would you handle this problem if the angles were expressed differently?

Tip: When solving for unknowns in geometric problems, always start by using known formulas or relationships (like the sum of interior angles) to set up the equation.