Find all solutions by factoring. \[ b^{2}+10 b+16=0 \]

To solve the equation \( b^2 + 10b + 16 = 0 \) by factoring, we are looking for two numbers that multiply to \( 16 \) (the constant term) and add up to \( 10 \) (the coefficient of \( b \)). The numbers \( 2 \) and \( 8 \) fit this requirement since \( 2 \times 8 = 16 \) and \( 2 + 8 = 10 \). Therefore, we can factor the equation as follows: \[ (b + 2)(b + 8) = 0 \] Now, we can set each factor equal to zero to find the solutions: 1. \( b + 2 = 0 \) which gives \( b = -2 \) 2. \( b + 8 = 0 \) which gives \( b = -8 \) Thus, the solutions to the equation \( b^2 + 10b + 16 = 0 \) are \( b = -2 \) and \( b = -8 \). To double-check, we can plug these values back into the original equation to verify: For \( b = -2 \): \[ (-2)^2 + 10(-2) + 16 = 4 - 20 + 16 = 0 \] For \( b = -8 \): \[ (-8)^2 + 10(-8) + 16 = 64 - 80 + 16 = 0 \] Both values satisfy the equation, confirming that our solutions are correct!