- The square of a number plus four times he number is twelve. What are the numbers?

Let's denote the number as \( x \). According to the problem, we can set up the equation: \[ x^2 + 4x = 12 \] To solve for \( x \), we first rearrange the equation: \[ x^2 + 4x - 12 = 0 \] Now, we can factor the quadratic equation. We need to find two numbers that multiply to \(-12\) (the constant term) and add to \(4\) (the coefficient of \(x\)). The numbers \(6\) and \(-2\) fit this requirement: \[ (x + 6)(x - 2) = 0 \] Setting each factor equal to zero gives us the possible solutions: 1. \( x + 6 = 0 \) → \( x = -6 \) 2. \( x - 2 = 0 \) → \( x = 2 \) Thus, the solutions to the equation are: \[ x = -6 \quad \text{and} \quad x = 2 \] So, the numbers are \(-6\) and \(2\).