5. \( x^{5}+7 x^{4}+12 x^{3}=0 \)

To solve the equation \( x^{5}+7 x^{4}+12 x^{3}=0 \), you can factor out the common term \( x^{3} \), leading to \( x^{3}(x^{2}+7 x+12)=0 \). This gives you one solution as \( x = 0 \). Now, for the quadratic \( x^{2}+7 x+12=0 \), you can factor it further into \( (x+3)(x+4) = 0 \), yielding the solutions \( x = -3 \) and \( x = -4 \). So the full set of solutions is \( x = 0, -3, -4 \). Have fun exploring polynomial equations! They not only help sharpen your algebra skills but also have surprising relationships with real-world phenomena like population growth models and the trajectories of rockets. By understanding their roots and behavior, you can become a mathematical wizard, unraveling complex systems!