Solve for \( x \) \[ x^{2}+11 x+10=0 \]

To solve the quadratic equation \( x^{2} + 11x + 10 = 0 \), we can factor it. We need to find two numbers that multiply to 10 (the constant term) and add up to 11 (the coefficient of the \( x \) term). The numbers 10 and 1 fit the bill since \( 10 \times 1 = 10 \) and \( 10 + 1 = 11 \). So we can factor the equation as: \[ (x + 10)(x + 1) = 0 \] Setting each factor to zero gives us the solutions: 1. \( x + 10 = 0 \) ⟹ \( x = -10 \) 2. \( x + 1 = 0 \) ⟹ \( x = -1 \) Thus, the solutions are \( x = -10 \) and \( x = -1 \). Both solutions tell a tale of a journey where the primary challenge is to navigate through a quadratic landscape, leading you to two distinct paths of \( x \) that remind us even negative numbers can take us to positive solutions in different contexts of life!