In this image, we are given that $m \parallel n$, meaning that the two lines are parallel. Two angles are provided: $(4x - 6)^\circ$ and $(4x + 10)^\circ$.

Since the lines $m$ and $n$ are parallel, the angles shown are corresponding angles. Corresponding angles are equal when two lines are parallel, so we can set up the following equation:

$4x - 6 = 4x + 10$

Now, let's solve this equation for $x$:

1. Subtract $4x$ from both sides: $-6 = 10$

However, this results in a contradiction, indicating that these two angles are not equal. This means that they are actually supplementary angles (they add up to 180°), which is the case for interior angles formed by parallel lines and a transversal.

Thus, we revise the equation to:

$(4x - 6) + (4x + 10) = 180$

Simplify the equation:

$8x + 4 = 180$

Now, solve for $x$:

1. Subtract 4 from both sides: $8x = 176$

2. Divide by 8: $x = 22$

The value of $x$ is $22$.

Would you like further details on any part of this solution? Here are five related questions you can explore:

1. What are corresponding angles, and how are they used in geometry?
2. Why are the angles in this problem considered supplementary?
3. What is the transversal line, and how does it relate to parallel lines?
4. How can we determine if angles are equal or supplementary in parallel line problems?
5. How do the properties of parallel lines apply to real-world problems?

Tip: When working with angles and parallel lines, always check whether the angles are corresponding, alternate interior, or supplementary to set up the correct equation.