The diagram you've provided shows a circle with a point $V$ at its center. Two line segments extend from the circle: one from point $A$ on the circle to $Z$, and another from point $B_1$ on the circle to $Z$. The lengths of these segments are given as $6x + 20$ and $10x + 4$, respectively.

Since both $A$ and $B_1$ are points on the circle and $Z$ lies outside the circle, these two lines are tangent segments to the circle from the same external point $Z$. According to the tangent-segment theorem, the two tangents from a common external point to a circle are equal in length.

Thus, we can set the two expressions for the lengths equal to each other:

$6x + 20 = 10x + 4$

Solving for $x$:

1. Subtract $6x$ from both sides: $20 = 4x + 4$

2. Subtract 4 from both sides: $16 = 4x$

3. Divide both sides by 4: $x = 4$

So, the value of $x$ is $\boxed{4}$.

Would you like more details or have any questions? Here are some related questions:

1. What is the tangent-segment theorem, and why does it apply here?
2. How can you verify that $x = 4$ satisfies the given condition geometrically?
3. What is the general formula for finding the length of tangents from a point outside a circle?
4. How do the angles formed by tangents relate to the center of the circle?
5. How would the problem change if point $Z$ were inside the circle?

Tip: Always check that your solution satisfies all geometric properties in a problem, especially when tangents are involved.