1. **State the problem:** We are given two complex numbers $z = -17 - 6i$ and $w = 3 + 1i$. We need to find the value of $u$ that satisfies the equation: $$\frac{u}{10u} = \frac{3 + 1i}{z + 2w}$$ 2. **Rewrite the equation:** $$\frac{u}{10u} = \frac{3 + i}{z + 2w}$$ 3. **Simplify the left side:** Since $u \neq 0$, we can cancel $u$: $$\frac{\cancel{u}}{10\cancel{u}} = \frac{1}{10}$$ So the equation becomes: $$\frac{1}{10} = \frac{3 + i}{z + 2w}$$ 4. **Calculate $z + 2w$:** $$z + 2w = (-17 - 6i) + 2(3 + i) = -17 - 6i + 6 + 2i = (-17 + 6) + (-6i + 2i) = -11 - 4i$$ 5. **Set up the equation:** $$\frac{1}{10} = \frac{3 + i}{-11 - 4i}$$ 6. **Cross multiply:** $$-11 - 4i = 10(3 + i)$$ 7. **Calculate the right side:** $$10(3 + i) = 30 + 10i$$ 8. **Check equality:** $$-11 - 4i = 30 + 10i$$ This is not true, so the initial assumption that $u \neq 0$ and canceling $u$ directly might be incorrect. Let's solve for $u$ explicitly. 9. **Rewrite original equation:** $$\frac{u}{10u} = \frac{3 + i}{z + 2w}$$ Simplify left side: $$\frac{u}{10u} = \frac{1}{10}$$ So the equation is: $$\frac{1}{10} = \frac{3 + i}{z + 2w}$$ Cross multiply: $$z + 2w = 10(3 + i)$$ Calculate right side: $$10(3 + i) = 30 + 10i$$ Calculate left side: $$z + 2w = -11 - 4i$$ Since $-11 - 4i \neq 30 + 10i$, the equation cannot hold unless $u=0$ which would make the left side undefined. 10. **Conclusion:** The only way for the equation to hold is if $u=0$, but then the left side is undefined. Therefore, no value of $u$ satisfies the equation. **Final answer:** $$\boxed{\text{No solution for } u}$$