METHOD I: FACTORING 1. \( x^{2}-6 x-40=0 \) \[ 2\left(x^{2}-3 x-20\right) \]

**Step 1. Write Down the Equation** We start with the quadratic equation: \[ x^2-6x-40=0 \] **Step 2. Find Two Numbers** We need to find two numbers that multiply to \(-40\) (the constant term) and add to \(-6\) (the coefficient of \(x\)). After testing a few pairs, we find that the numbers \(4\) and \(-10\) work because: \[ 4 \times (-10) = -40 \quad \text{and} \quad 4 + (-10) = -6 \] **Step 3. Factor the Quadratic** Using these numbers, we can factor the quadratic as follows: \[ x^2-6x-40=(x+4)(x-10) \] **Step 4. Set Each Factor Equal to Zero** For the product to be zero, at least one of the factors must equal zero. Thus, we set: \[ x+4=0 \quad \text{or} \quad x-10=0 \] **Step 5. Solve for \(x\)** Solving each equation gives: For \(x+4=0\): \[ x = -4 \] For \(x-10=0\): \[ x = 10 \] **Conclusion** The solutions to the equation are: \[ x=-4 \quad \text{or} \quad x=10 \]