d (1-2 x)/(2 x)+(2 x)/(2 x-1)+(1)/(2 x-4 x^2)

1. **Factor out common factors:** <br /> - The denominator of the third term can be factored: $2x - 4x^2 = 2x(1-2x)$<br /><br />2. **Find a common denominator:**<br /> - The least common denominator is $2x(1-2x)$.<br /><br />3. **Rewrite each fraction with the common denominator:**<br /> - $\frac{1-2x}{2x} \cdot \frac{1-2x}{1-2x} = \frac{(1-2x)^2}{2x(1-2x)}$<br /> - $\frac{2x}{2x-1} \cdot \frac{2x}{2x} = \frac{4x^2}{2x(1-2x)}$<br /> - $\frac{1}{2x(1-2x)}$ (already has the common denominator)<br /><br />4. **Combine the numerators:**<br /> - $\frac{(1-2x)^2 + 4x^2 + 1}{2x(1-2x)}$<br /><br />5. **Simplify the numerator:**<br /> - $\frac{1 - 4x + 4x^2 + 4x^2 + 1}{2x(1-2x)}$<br /> - $\frac{8x^2 - 4x + 2}{2x(1-2x)}$<br /><br />6. **Factor out a 2 from the numerator:**<br /> - $\frac{2(4x^2 - 2x + 1)}{2x(1-2x)}$<br /><br />7. **Cancel the common factor of 2:**<br /> - $\frac{4x^2 - 2x + 1}{x(1-2x)}$<br /><br />**Therefore, the simplified expression is $\frac{4x^2 - 2x + 1}{x(1-2x)}$.** <br />