To solve for $x$ in this problem, we need to recognize that the shape shown is a quadrilateral. The sum of the internal angles in any quadrilateral is $360^\circ$.

Each angle in the quadrilateral is given in terms of $x$:

1. $(7x + 4)^\circ$
2. $(5x + 4)^\circ$
3. $(4x + 9)^\circ$
4. $(9x - 6)^\circ$

Adding these together and setting the sum equal to $360^\circ$, we get:

$(7x + 4) + (5x + 4) + (4x + 9) + (9x - 6) = 360$

Combine like terms:

$7x + 5x + 4x + 9x + 4 + 4 + 9 - 6 = 360$

$25x + 11 = 360$

Subtract 11 from both sides:

$25x = 349$

Now, divide by 25:

$x = \frac{349}{25} = 13.96$

Since the answer choices provided are rounded whole numbers, the closest answer to $x = 13.96$ is $14$.

Thus, the solution is: $x \approx 14$

Would you like further details on this solution, or do you have any questions?

Here are five related questions:

1. What is the sum of the interior angles for a pentagon?
2. How do you determine the sum of angles in any polygon?
3. If one angle is given in degrees and the others in terms of $x$, how does that affect the equation setup?
4. How can we confirm that the answer choice rounding was appropriate in this case?
5. What are other types of problems involving angle sums in polygons?

Tip: Always remember to verify the sum of angles based on the polygon's shape type; for quadrilaterals, it is always $360^\circ$.