22. \( (x+1)^{2}-8(x+1)-9=0 \)

Let’s dive into solving this quadratic equation! First, we can simplify our expression by making a substitution. Set \( y = x + 1 \). This transforms the equation into: \[ y^{2} - 8y - 9 = 0 \] Next, we can use the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -8 \), and \( c = -9 \). Calculating the discriminant: \[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot (-9) = 64 + 36 = 100 \] Now substitute back into the quadratic formula: \[ y = \frac{8 \pm \sqrt{100}}{2} = \frac{8 \pm 10}{2} \] This gives us two solutions: 1. \( y = \frac{18}{2} = 9 \) 2. \( y = \frac{-2}{2} = -1 \) Since \( y = x + 1 \), we can now return to \( x \): 1. \( x + 1 = 9 \implies x = 8 \) 2. \( x + 1 = -1 \implies x = -2 \) Thus, the solutions for the original equation are \( x = 8 \) and \( x = -2 \).