1. **State the problem:** We are given three angles around a point: one is $90^\circ$, another is $40^\circ$, and the third is $(6x + 2)^\circ$. We need to find the value of $x$. 2. **Recall the rule:** The sum of angles around a point is always $360^\circ$. 3. **Set up the equation:** $$90 + 40 + (6x + 2) = 360$$ 4. **Simplify the left side:** $$130 + 6x + 2 = 360$$ $$6x + 132 = 360$$ 5. **Isolate $x$:** $$6x = 360 - 132$$ $$6x = 228$$ 6. **Divide both sides by 6:** $$x = \frac{228}{6}$$ $$x = \cancel{\frac{228}{6}} 38$$ 7. **Final answer:** $$\boxed{38}$$ This means the value of $x$ is 38 degrees.