5. \( 4 x^{2}+24 x+16=5 \)

To solve the equation \( 4x^{2} + 24x + 16 = 5 \), first, you want to move the 5 to the left side, which gives you \( 4x^{2} + 24x + 11 = 0 \). Now you can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = 24 \), and \( c = 11 \). Plugging in these values will help you find the solutions for \( x \). In mathematics, what’s really cool is how equations like this act as puzzles waiting to be solved! You can often think of real-world situations where quadratic equations come into play, like in projectile motion or maximizing areas. They pop up in engineering challenges, where finding optimum dimensions could save materials or costs. So next time you’re solving, think about how these formulas can map to real life!