\( \frac { x ^ { 2 } + 5 x } { k ^ { 2 } + 16 } > 0 \)

To solve the inequality \[ \frac{x^2 + 5x}{k^2 + 16} > 0, \] we need to analyze the expression on the left side. 1. **Denominator Analysis**: The denominator \(k^2 + 16\) is always positive for all real values of \(k\) because \(k^2\) is non-negative and adding 16 ensures that it is strictly greater than zero. Therefore, we can focus on the numerator. 2. **Numerator Analysis**: The numerator is \(x^2 + 5x\). We can factor this expression: \[ x^2 + 5x = x(x + 5). \] 3. **Finding Roots**: The roots of the numerator are found by setting it equal to zero: \[ x(x + 5) = 0. \] This gives us the roots \(x = 0\) and \(x = -5\). 4. **Sign Analysis**: We will analyze the sign of the expression \(x(x + 5)\) in the intervals determined by the roots: - **Interval 1**: \(x < -5\) - Choose \(x = -6\): \[ -6(-6 + 5) = -6(-1) = 6 > 0. \] - **Interval 2**: \(-5 < x < 0\) - Choose \(x = -1\): \[ -1(-1 + 5) = -1(4) = -4 < 0. \] - **Interval 3**: \(x > 0\) - Choose \(x = 1\): \[ 1(1 + 5) = 1(6) = 6 > 0. \] 5. **Conclusion**: The expression \(x(x + 5)\) is positive in the intervals \( (-\infty, -5) \) and \( (0, \infty) \), and it is negative in the interval \((-5, 0)\). The expression is zero at \(x = -5\) and \(x = 0\). Thus, the solution to the inequality \[ \frac{x^2 + 5x}{k^2 + 16} > 0 \] is: \[ x \in (-\infty, -5) \cup (0, \infty). \]