Reduce the rational expression to lowest terms. \( \frac{4 s^{2}+28 s+48}{16 s^{2}-32 s-240} \) \( \frac{4 s^{2}+28 s+48}{16 s^{2}-32 s-240}=\square \)

To reduce the rational expression \(\frac{4 s^{2}+28 s+48}{16 s^{2}-32 s-240}\) to its lowest terms, begin by factoring both the numerator and denominator. For the numerator \(4s^{2} + 28s + 48\), you can factor out a 4: \[ 4(s^{2} + 7s + 12) \] Now factor \(s^{2} + 7s + 12\): \[ s^{2} + 7s + 12 = (s + 3)(s + 4) \] So, the factored form of the numerator is: \[ 4(s + 3)(s + 4) \] For the denominator \(16s^{2} - 32s - 240\), factor out a 16: \[ 16(s^{2} - 2s - 15) \] Now factor \(s^{2} - 2s - 15\): \[ s^{2} - 2s - 15 = (s - 5)(s + 3) \] So, the factored form of the denominator is: \[ 16(s - 5)(s + 3) \] Now you can rewrite the entire expression as: \[ \frac{4(s + 3)(s + 4)}{16(s - 5)(s + 3)} \] Next, cancel the common factor \((s + 3)\): \[ \frac{4(s + 4)}{16(s - 5)} \] Finally, simplify: \[ \frac{s + 4}{4(s - 5)} \] So, the reduced form is \(\frac{s + 4}{4(s - 5)}\).