Select the correct answer. Solve the system of equations. \( \begin{array}{l}y=4 x+5 \\ y=x^{2}+12 x+20 \\ \text { A. } \quad(3,17) \text { and }(5,-15) \\ \text { B. } \quad(-3,-7) \text { and }(-5,-15) \\ \text { C. } \quad(-3,-15) \text { and }(-5,-7) \\ \text { D. } \quad(-3,-17) \text { and }(-5,-25)\end{array} \)

To solve the system of equations given by \( y = 4x + 5 \) and \( y = x^2 + 12x + 20 \), we can set them equal to each other since both equal \( y \): \[ 4x + 5 = x^2 + 12x + 20. \] Rearranging gives us: \[ 0 = x^2 + 12x + 20 - 4x - 5, \] which simplifies to: \[ 0 = x^2 + 8x + 15. \] Factoring this quadratic yields: \[ (x + 3)(x + 5) = 0, \] leading to the solutions \( x = -3 \) and \( x = -5 \). Plugging these values back into the linear equation \( y = 4x + 5 \): - For \( x = -3 \): \( y = 4(-3) + 5 = -12 + 5 = -7 \) - For \( x = -5 \): \( y = 4(-5) + 5 = -20 + 5 = -15 \) So, the points of intersection are \( (-3, -7) \) and \( (-5, -15) \). Checking the options, the correct answer is: **B. \((-3, -7)\) and \((-5, -15)\)**. Isn't it fascinating how two seemingly different equations can intersect at distinct points? These solutions also serve as crucial data in various fields such as physics and engineering, where understanding intersections can represent optimal solutions or critical points in a system's behavior. If you're eager to dive deeper into solving quadratic equations or systems of equations, consider exploring resources that cover the quadratic formula, graphing techniques, and numerical methods. These can enrich your problem-solving toolkit while enhancing your mathematical understanding!