Factor each completely \( \therefore \quad \times \quad b^{2}-6 b+8 \)

To factor the expression \( b^{2}-6b+8 \), we look for two numbers that multiply to \( 8 \) (the constant term) and add to \( -6 \) (the coefficient of \( b \)). The numbers \( -2 \) and \( -4 \) fit this requirement since \( -2 \times -4 = 8 \) and \( -2 + (-4) = -6 \). Therefore, we can express the quadratic as: \[ (b - 2)(b - 4) \] Next, let’s see this operation in a real-world context. Quadratics like this one often arise in scenarios like maximizing area. Imagine you’re designing a rectangular garden with a fixed area. The factors \( b - 2 \) and \( b - 4 \) can represent the lengths of the sides where the total area is determined by \( 8 \) square units. This shows how algebra can apply to real-life planning! A common mistake when factoring quadratics is forgetting to check your factors. After factoring, you should always remember to expand back to verify your answer: \( (b - 2)(b - 4) \) indeed gives us \( b^2 - 6b + 8 \). Skipping this verification step can lead to incorrect answers—so double-check your work!