Question Solve the equation for all values of \( x \). \( -2 x(x+7)\left(x^{2}-11\right)=0 \) Answer Attempt 1 out of 2 \( \oplus \) Additional Solution \( \Theta \) No Solution

To solve the equation \( -2 x(x+7)(x^{2}-11)=0 \), we can break it down by setting each factor equal to zero. The first factor gives \( x=0 \). The second factor \( x+7=0 \) leads to \( x=-7 \). The third factor \( x^{2}-11=0 \) can be solved to find \( x=\sqrt{11} \) and \( x=-\sqrt{11} \). Therefore, the solutions are \( x=0, x=-7, x=\sqrt{11}, \) and \( x=-\sqrt{11} \). This equation not only illustrates how polynomials can be factored and solved, but also ties into many real-life scenarios, like analyzing projectile motion or optimizing areas in geometry. Whenever you encounter polynomial equations in real life, like calculating dimensions of a structure or understanding growth models, you’re essentially doing what you just did: breaking down the problem into manageable parts. Fun fact: these equations can sometimes describe everything from how a ball bounces to the trajectory of a rocket!