Factor the trinomial completely. If the trinomial cannot be factored, say it is prime. \[ r^{2}+r s-6 s^{2} \]

To factor the trinomial \( r^{2}+rs-6s^{2} \), we look for two numbers that multiply to give the product of the leading coefficient (1) and the constant term (-6s²), which is -6s², and add up to the coefficient of the middle term (1). The numbers that meet these requirements are 3s and -2s. Thus, we can rewrite the middle term: \[ r^{2} + 3rs - 2rs - 6s^{2} \] Next, we group the terms: \[ (r^{2} + 3rs) + (-2rs - 6s^{2}) \] Now we can factor by grouping: \[ r(r + 3s) - 2s(r + 3s) \] Factoring out the common binomial \((r + 3s)\): \[ (r + 3s)(r - 2s) \] So the factored form of the trinomial \( r^{2}+rs-6s^{2} \) is \((r + 3s)(r - 2s)\).