Solve by factoring. \[ 5 x^{2}-25 x-120=0 \]

To solve the equation \( 5x^{2}-25x-120=0 \) by factoring, we first factor out the greatest common factor, which is 5: \[ 5(x^{2} - 5x - 24) = 0. \] Now, we can focus on factoring the quadratic \( x^{2} - 5x - 24 \). We need two numbers that multiply to -24 (the constant term) and add to -5 (the linear coefficient). The numbers -8 and 3 fit the bill since: \[ -8 \cdot 3 = -24 \quad \text{and} \quad -8 + 3 = -5. \] So, we can write the quadratic as: \[ (x - 8)(x + 3) = 0. \] Now, we can set each factor equal to zero: 1. \( x - 8 = 0 \) gives \( x = 8 \). 2. \( x + 3 = 0 \) gives \( x = -3 \). Thus, the solutions are: \[ x = 8 \quad \text{and} \quad x = -3. \]