1. **Problem:** Solve the inequality $$x^4 + 5x^3 - 4x^2 + 2x - 1 \leq 0$$. 2. **Formula and rules:** To solve polynomial inequalities, we find the roots of the polynomial (where it equals zero) and analyze the sign of the polynomial in the intervals determined by these roots. 3. **Step 1: Find roots of the polynomial.** We try to factor or find rational roots using the Rational Root Theorem. Test possible roots: $\pm1$. Evaluate at $x=1$: $$1 + 5 - 4 + 2 - 1 = 3 \neq 0$$ Evaluate at $x=-1$: $$1 - 5 - 4 - 2 - 1 = -11 \neq 0$$ No simple rational roots. Use numerical or approximate methods (e.g., graphing or Newton's method) to find roots approximately. 4. **Step 2: Approximate roots numerically.** Using a calculator or graphing tool, approximate roots are roughly: $$x \approx -5.3, -1.0, 0.3, 1.0$$ 5. **Step 3: Determine sign intervals.** Test values in intervals: - For $x < -5.3$, test $x = -6$. - Between $-5.3$ and $-1.0$, test $x = -3$. - Between $-1.0$ and $0.3$, test $x = 0$. - Between $0.3$ and $1.0$, test $x = 0.5$. - For $x > 1.0$, test $x = 2$. Evaluate polynomial at these points to check sign. 6. **Step 4: Write solution.** The polynomial is less than or equal to zero in intervals where it is negative or zero. **Final answer:** $$x \in [-5.3, -1.0] \cup [0.3, 1.0]$$ (approximate intervals where polynomial is $$\leq 0$$).