1. **State the problem:** Simplify or analyze the expression $4x^4 - 33x^2 - 9x + 2$. 2. **Identify the expression:** This is a polynomial of degree 4 with terms $4x^4$, $-33x^2$, $-9x$, and $+2$. 3. **Look for factoring possibilities:** Since it is a quartic polynomial, try to factor by grouping or use substitution if possible. 4. **Attempt substitution:** Let $y = x^2$, then the polynomial becomes $4y^2 - 33y - 9x + 2$, but the $-9x$ term prevents direct substitution. 5. **Try factoring by grouping:** Group terms as $(4x^4 - 33x^2) + (-9x + 2)$. 6. **Factor out common terms:** From the first group, factor out $x^2$: $x^2(4x^2 - 33)$; the second group $-9x + 2$ has no common factor. 7. **No common binomial factor:** Since the groups do not share a common factor, factoring by grouping is not straightforward. 8. **Check for rational roots using Rational Root Theorem:** Possible roots are factors of 2 over factors of 4: $\pm1, \pm\frac{1}{2}, \pm2, \pm\frac{1}{4}$. 9. **Test $x=1$:** $4(1)^4 - 33(1)^2 - 9(1) + 2 = 4 - 33 - 9 + 2 = -36$ (not zero). 10. **Test $x=2$:** $4(16) - 33(4) - 18 + 2 = 64 - 132 - 18 + 2 = -84$ (not zero). 11. **Test $x=\frac{1}{2}$:** $4(\frac{1}{16}) - 33(\frac{1}{4}) - 9(\frac{1}{2}) + 2 = \frac{1}{4} - 8.25 - 4.5 + 2 = -10.5$ (not zero). 12. **No simple rational roots found; polynomial likely irreducible over rationals.** 13. **Summary:** The polynomial $4x^4 - 33x^2 - 9x + 2$ does not factor easily with rational roots or simple methods. **Final answer:** The expression is already in its simplest polynomial form.