a) \( \frac{1}{x+1}-\frac{2}{x^{2}-x+1} \leq \frac{1-2 x}{x^{3}+1} \)

To solve the inequality \( \frac{1}{x+1}-\frac{2}{x^{2}-x+1} \leq \frac{1-2 x}{x^{3}+1} \), we first express all terms with a common denominator to simplify our comparison. The denominators present are \( x+1 \), \( x^2 - x + 1 \), and \( x^3 + 1 \). Notably, \( x^3 + 1 \) can be factored as \( (x+1)(x^2 - x + 1) \). Thus, we can combine the fractions. 1. **Multiply through by the common denominator** (assuming \( x + 1 \neq 0 \) and \( x^2 - x + 1 \neq 0 \) to avoid undefined expressions): \[ (x^2 - x + 1)(x^3 + 1) \left(\frac{1}{x + 1} - \frac{2}{x^2 - x + 1}\right) \leq (1 - 2x)(x^2 - x + 1) \] 2. **Simplify the left side**: Simplifying \( \frac{1}{x+1} - \frac{2}{x^2-x+1} \) involves finding \( (x^2 - x + 1) - 2(x + 1) \) through proper arrangements. 3. **Rearranging yields**: Continuing along, we find that solving such inequalities often leads into polynomial expressions, as we look for values of \( x \) that satisfy our condition. 4. **Critical Points**: Next, we need to check these critical points where the numerator equals zero or where the fractions become undefined. You may factor or use synthetic division depending on the order of the polynomial. 5. **Test Intervals**: Finally, once we've identified critical points on the number line, we test intervals between these points to see if the original inequality holds. It's essential to plug numbers from each interval back into the inequality to verify. In conclusion, we find valid solutions based on our intervals of positivity/negativity stemming from the critical points derived from clearing the denominators. This logical breakdown leads us to a comprehensive understanding of potential values for \( x \) that would satisfy the original inequality. Keep in mind, visualizing this process with graphs might also help illuminate how these varied expressions intersect.