1. **State the problem:** We are given the function $w(x) = -12x^{5} + 12x^{3} - 3x$ and we want to analyze or simplify it as needed. 2. **Identify the type of function:** This is a polynomial function of degree 5. 3. **Look for common factors:** Notice each term has a factor of $-3x$. 4. **Factor out the common factor:** $$w(x) = -3x(4x^{4} - 4x^{2} + 1)$$ 5. **Check if the quartic inside can be factored further:** Let $y = x^{2}$, then the expression inside parentheses becomes: $$4y^{2} - 4y + 1$$ 6. **Factor the quadratic in $y$:** $$4y^{2} - 4y + 1 = (2y - 1)^{2}$$ 7. **Rewrite the factorization:** $$w(x) = -3x(2x^{2} - 1)^{2}$$ **Final answer:** $$w(x) = -3x(2x^{2} - 1)^{2}$$ This factorization shows the structure of the polynomial clearly and can be useful for solving equations or analyzing the function.