To solve this problem, we need to determine the value of $x$ that makes lines $\ell$ and $m$ parallel.

The two given angles are:

1. $(2x + 6)^\circ$, which is an angle on line $\ell$,
2. $130^\circ$, which is an angle on line $m$.

These two angles are corresponding angles because they are positioned similarly relative to the transversal cutting through the two lines. For lines $\ell$ and $m$ to be parallel, the corresponding angles must be equal.

Thus, we set the two angles equal to each other: $2x + 6 = 130$

Solving for $x$:

1. Subtract 6 from both sides: $2x = 130 - 6$ $2x = 124$

2. Divide both sides by 2: $x = \frac{124}{2}$ $x = 62$

Thus, the value of $x$ is $\boxed{62}$.

Would you like further clarification on any part of this solution? Here are some related questions to consider:

1. What are corresponding angles, and how are they related to parallel lines?
2. Could this problem be solved using alternate interior angles instead of corresponding angles?
3. How do you verify if lines are parallel based on angle relationships?
4. What other angle pairs can prove two lines are parallel?
5. How does this concept of parallel lines extend to geometry problems involving transversals and polygons?

Tip: Remember, when dealing with parallel lines cut by a transversal, corresponding, alternate interior, and consecutive interior angles can help determine relationships between angles!