1. **State the problem:** Solve the equation $$1 + 4x \sin\left(\frac{1}{x}\right) - 2 \cos\left(\frac{1}{x}\right) = 0.$$\n\n2. **Rewrite the equation:** We want to find $x$ such that $$1 + 4x \sin\left(\frac{1}{x}\right) - 2 \cos\left(\frac{1}{x}\right) = 0.$$\n\n3. **Isolate terms:** Move constants to the other side: $$4x \sin\left(\frac{1}{x}\right) = 2 \cos\left(\frac{1}{x}\right) - 1.$$\n\n4. **Divide both sides by 4:** $$x \sin\left(\frac{1}{x}\right) = \frac{2 \cos\left(\frac{1}{x}\right) - 1}{4}.$$\n\n5. **Substitute $t = \frac{1}{x}$:** Then $x = \frac{1}{t}$ and the equation becomes $$\frac{1}{t} \sin(t) = \frac{2 \cos(t) - 1}{4}.$$\n\n6. **Multiply both sides by $t$:** $$\sin(t) = \frac{t}{4} (2 \cos(t) - 1).$$\n\n7. **Rewrite:** $$\sin(t) = \frac{t}{4} (2 \cos(t) - 1).$$\n\n8. **Solve for $t$ numerically:** This transcendental equation cannot be solved exactly by elementary methods. We look for values of $t$ satisfying $$\sin(t) - \frac{t}{4} (2 \cos(t) - 1) = 0.$$\n\n9. **Recall $x = \frac{1}{t}$:** Once $t$ is found, $x = \frac{1}{t}$.\n\n**Summary:** The solutions $x$ satisfy $$1 + 4x \sin\left(\frac{1}{x}\right) - 2 \cos\left(\frac{1}{x}\right) = 0,$$ equivalently $$\sin(t) = \frac{t}{4} (2 \cos(t) - 1)$$ with $t = \frac{1}{x}$. Numerical methods are needed to find approximate roots for $t$, then invert to find $x$.