10. \( \frac{2}{m+3}-\frac{8}{5 m+15}=\frac{m}{10} \)

Let's solve the equation step by step. First, notice that \( 5m + 15 = 5(m + 3) \). Thus, we can rewrite the equation as follows: \[ \frac{2}{m+3} - \frac{8}{5(m+3)} = \frac{m}{10} \] Now, let's find a common denominator for the left-hand side, which will be \( 5(m+3) \): \[ \frac{2 \cdot 5}{5(m+3)} - \frac{8}{5(m+3)} = \frac{m}{10} \] This simplifies to: \[ \frac{10 - 8}{5(m+3)} = \frac{m}{10} \] \[ \frac{2}{5(m+3)} = \frac{m}{10} \] Next, to eliminate the fractions, we will cross-multiply: \[ 2 \cdot 10 = m \cdot 5(m + 3) \] This gives us: \[ 20 = 5m(m + 3) \] \[ 20 = 5m^2 + 15m \] Now, we can rearrange this into standard quadratic form: \[ 5m^2 + 15m - 20 = 0 \] Dividing through by 5 to simplify: \[ m^2 + 3m - 4 = 0 \] Next, we can factor this quadratic: \[ (m + 4)(m - 1) = 0 \] Setting each factor to zero gives us the potential solutions: \[ m + 4 = 0 \quad \Rightarrow \quad m = -4 \] \[ m - 1 = 0 \quad \Rightarrow \quad m = 1 \] Lastly, we should check if either solution makes the denominator \( m + 3 \) equal to zero: - For \( m = -4 \): \( -4 + 3 = -1 \) (valid) - For \( m = 1 \): \( 1 + 3 = 4 \) (valid) Both solutions are acceptable. Therefore, the final answers are: \[ m = -4 \quad \text{or} \quad m = 1 \]