To solve the problem, let's analyze the provided diagram and the given conditions:

Given:

1. $MNOP$ is a parallelogram.
2. $NO = NQ$.
3. $\angle M = 110^\circ$.
4. $\angle N_2 = 4x$.

To find:

The value of $x$.

Solution:

1. Understanding the Angles:

   • Since $MNOP$ is a parallelogram, opposite angles are equal. Therefore: $\angle M = \angle P = 110^\circ$
   • The sum of the interior angles of a parallelogram is $360^\circ$, and opposite angles are equal, so: $\angle N + \angle O = 360^\circ - 2(110^\circ) = 360^\circ - 220^\circ = 140^\circ$
   • Therefore: $\angle N + \angle O = 140^\circ$

2. Examining $\angle N$ and $\angle O$:

   • The given problem indicates that $\angle N_2 = 4x$. Since $NQ = NO$, triangle $NQO$ is isosceles, and the angles $\angle N_1$ and $\angle O_1$ are equal. Let's call these angles $y$.
   • Therefore: $\angle N_1 = \angle O_1 = y$
   • Since $\angle N = \angle N_1 + \angle N_2$ and $\angle O = 2y$, we can substitute these values into the sum: $y + 4x + 2y = 140^\circ$ Simplifying: $3y + 4x = 140^\circ$

3. Using the Parallel Lines:

   • Since $NO = NQ$, $\angle N_1$ and $\angle O_1$ are congruent, and by the property of the parallelogram and the isosceles triangle $NQO$, $\angle O_1 = \angle O_2 = y$.

4. Solving for $x$:

   • To find $y$, observe that $\angle M + \angle N$ must equal $180^\circ$ (since they are consecutive angles in a parallelogram): $110^\circ + 4x = 180^\circ$
   • Subtracting $110^\circ$ from both sides: $4x = 70^\circ$
   • Dividing by 4: $x = 17.5^\circ$

Final Answer:

The value of $x$ is $17.5^\circ$.

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

1. What are the properties of angles in a parallelogram?
2. How would the problem change if $\angle M$ was different?
3. Can we determine the length of the sides if only angles are given?
4. What happens to the values if $NQ$ is not equal to $NO$?
5. How do the properties of isosceles triangles help solve this problem?

Tip: Remember, in geometry, always use the properties of shapes to relate angles and sides, as this can often simplify complex problems.