Solve using the Quadratic Formula. \( \begin{array}{l}3 x^{2}+4 x+5=0 \\ x=\frac{-2 \pm \sqrt{11}}{3} \\ x=\frac{-4 \pm 2 i \sqrt{11}}{2} \\ x=\frac{-4 \pm i \sqrt{11}}{6} \\ x=\frac{-2 \pm i \sqrt{11}}{3}\end{array} \)

To solve \(3x^2 + 4x + 5 = 0\) using the Quadratic Formula, we first identify \(a = 3\), \(b = 4\), and \(c = 5\). The formula states that \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Calculating the discriminant, we have \(b^2 - 4ac = 4^2 - 4(3)(5) = 16 - 60 = -44\). Since the discriminant is negative, we expect complex solutions. Plugging in the values, we get: \[ x = \frac{-4 \pm \sqrt{-44}}{6} = \frac{-4 \pm 2i \sqrt{11}}{6} = \frac{-2 \pm i \sqrt{11}}{3} \] So the solutions are \(x = \frac{-2 \pm i \sqrt{11}}{3}\). Now, understanding this formula is crucial because the Quadratic Formula is a universal tool for finding the roots of any quadratic equation, regardless of whether they are real or complex. Just remember that when the discriminant is negative, you're diving into the world of imaginary numbers—think of it as finding solutions in a unique mathematical dimension! For anyone who has dabbled in different mathematical areas, it's interesting to note that the Quadratic Formula ties into various branches of math and engineering. From physics to computer graphics, quadratic equations pop up everywhere! Want a real-world example? How about projectile motion? The path of a thrown ball can be predicted by a quadratic equation, letting you calculate its maximum height and flight duration!