(4) \( x^{2}+8 x+15 \)

To truly appreciate the quadratic expression \( x^{2}+8x+15 \), you may want to consider its historical context. Quadratics have played a significant role in mathematics since ancient civilizations. Babylonians used geometric methods that resembled solving quadratic equations over 4,000 years ago! The development of algebra further formalized these methods, with the Persian mathematician Al-Khwarizmi laying the groundwork for quadratic solutions, making the journey to our modern understanding both exciting and rich. Now, let’s dive into how to solve such quadratics in real-world scenarios! This expression represents a parabolic graph, which can model various physical systems, from projectile motion to profit maximization in business. By finding the roots, or where the parabola crosses the x-axis, we can identify critical points—like when an object hits the ground or when profits break even. So, not only do these equations help us mathematically, but they also bring valuable insights into real-world situations!