Given the angles $m\angle NOP = 2x + 2$, $m\angle POR = 3x - 5$, and $m\angle NOQ = 144^\circ$, we are asked to find the value of $x$.

Assuming that points $N$, $O$, $P$, $Q$, and $R$ are on a straight line or form a linear pair (since $m\angle NOQ = 144^\circ$), we can deduce that the angles $m\angle NOP$ and $m\angle POR$ must sum up to $m\angle NOQ$. Thus, we have the equation:

$m\angle NOP + m\angle POR = m\angle NOQ$

Substituting the given expressions for the angles:

$(2x + 2) + (3x - 5) = 144$

Now, let's simplify and solve for $x$:

$2x + 3x + 2 - 5 = 144$

$5x - 3 = 144$

Add 3 to both sides:

$5x = 147$

Finally, divide by 5:

$x = \frac{147}{5} = 29.4$

So, the value of $x$ is 29.4.

Would you like a detailed explanation or have any questions?

Here are 5 related questions:

1. What is the measure of $m\angle NOP$ when $x = 29.4$?
2. How would you check if the angles sum correctly when $x = 29.4$?
3. Can the value of $x$ be negative in this scenario?
4. What assumptions are made about the relationship between points $N$, $O$, $P$, $Q$, and $R$?
5. How would the problem change if $m\angle NOQ$ was different?

Tip: Always verify your final answer by substituting it back into the original equations to ensure all conditions are met.