Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer, if necessar \( -3 u^{4}+27 u^{3}-45 u^{2}-75 u \geq 0 \) The solution set is \( \square . \square \square \square \square \square \square][\square, \square] \)

To solve the inequality \( -3u^{4}+27u^{3}-45u^{2}-75u \geq 0 \), we first factor it out. The expression can be factored as \( -3u(u^3 - 9u^2 + 15u + 25) \geq 0 \). Let’s find critical points by setting the equation to zero: Setting \( -3u = 0 \) gives \( u = 0 \). For the cubic polynomial \( u^3 - 9u^2 + 15u + 25 = 0 \), we can use numerical methods or graphing techniques to find its roots. Once we identify the roots, we will evaluate the intervals between these critical values to determine where the expression is positive. After solving the complete cubic, we might find the intervals such as \( (-\infty, -5] \), \( [0, 5] \) for our inequality. Thus, the solution set is \( (-\infty, -5] \cup [0, 5] \).