We are given the following information:

• $NP = 4x + 10$
• $NO = 4x$
• $OP = x + 5$

Since $O$ lies on the line segment $NP$, the sum of the lengths of $NO$ and $OP$ must equal the length of $NP$. This gives the equation:

$NO + OP = NP$

Substituting the given expressions:

$4x + (x + 5) = 4x + 10$

Now, simplify and solve for $x$:

$4x + x + 5 = 4x + 10$

$5x + 5 = 4x + 10$

Subtract $4x$ from both sides:

$x + 5 = 10$

Subtract 5 from both sides:

$x = 5$

Now, substitute $x = 5$ back into the expression for $NP$ to find the numerical length of $NP$:

$NP = 4x + 10 = 4(5) + 10 = 20 + 10 = 30$

Thus, the length of $NP$ is $\boxed{30}$.

Would you like further explanation or more details on any part? Here are 5 related questions:

1. How do you verify the position of a point on a line segment?
2. What other methods can be used to solve similar problems involving line segments?
3. How does the value of $x$ affect the lengths of $NO$ and $OP$?
4. What are some applications of this type of geometry problem in real-world situations?
5. How does the relationship between segment lengths help in geometric proofs?

Tip: When dealing with line segments and points in geometry, always ensure the sum of parts equals the whole segment length for consistent solutions.