1. **State the problem:** We are given the quadratic equation $3x^2 - 4x + 15 = 0$ and another quadratic $x^2 + bx + c = 0$ with the same roots. We need to find the ordered pair $(b,c)$. 2. **Recall the property of roots:** If two quadratics have the same roots, their roots' sum and product must be equal. 3. **Sum and product of roots for the first quadratic:** For $ax^2 + bx + c = 0$, sum of roots $= -\frac{b}{a}$ and product of roots $= \frac{c}{a}$. For $3x^2 - 4x + 15 = 0$, sum of roots $= -\frac{-4}{3} = \frac{4}{3}$ and product of roots $= \frac{15}{3} = 5$. 4. **Sum and product of roots for the second quadratic:** For $x^2 + bx + c = 0$, sum of roots $= -b$ and product of roots $= c$. 5. **Equate sums and products:** $$-b = \frac{4}{3} \implies b = -\frac{4}{3}$$ $$c = 5$$ 6. **Final answer:** The ordered pair is $$(b,c) = \left(-\frac{4}{3}, 5\right)$$