\( x^{2}+2 x-8=0 \) \( \left(x^{2}+4 x\right)(-2 x-8)=0 \) \( x(x+4)-2(x+4)=0 \) \( (x-2)(x-4)=0 \) \( x-2=0 \) or \( x+4=0 \) \( x-2 \) or \( x=4 \)

Let's dive into a lively exploration here! In the first equation, \( x^2 + 2x - 8 = 0 \), you're dealing with a quadratic. It has roots that can be derived using factors or the quadratic formula. It's fascinating how the structure of equations leads us toward solutions! Now, these solutions \( x = 2 \) and \( x = -4 \) mean that on a graph, the parabola intersects the x-axis at those points. This sets the stage for understanding how functions behave and apply in real life – think about projectile motion or profit models! Solving equations like these is the foundation for much more complex applications in physics and economics.