2. Identify the value(s) of \( m \) that satisfy the equation below, \( \quad \frac{4}{m-3}=\frac{m+5}{5} \)

To solve the equation \( \frac{4}{m-3}=\frac{m+5}{5} \), we'll first cross-multiply: \[ 4 \cdot 5 = (m - 3)(m + 5) \] This simplifies to: \[ 20 = m^2 + 5m - 3m - 15 \] or: \[ 20 = m^2 + 2m - 15 \] Next, let's rearrange the equation to set it equal to zero: \[ m^2 + 2m - 15 - 20 = 0 \] This becomes: \[ m^2 + 2m - 35 = 0 \] Now, we can factor the quadratic equation: \[ (m + 7)(m - 5) = 0 \] Setting each factor to zero gives us the solutions: \[ m + 7 = 0 \quad \Rightarrow \quad m = -7 \] \[ m - 5 = 0 \quad \Rightarrow \quad m = 5 \] Thus, the values of \( m \) that satisfy the equation are \( m = -7 \) and \( m = 5 \). Double-checking: substituting \( m = -7 \) into the original equation gives \( \frac{4}{-10} = -\frac{2}{5} \), which is true. Similarly, substituting \( m = 5 \) gives \( \frac{4}{2} = 2 \), also true. Therefore, both values are valid.