1. **State the problem:** Find the domain of the rational function $$F(x) = \frac{7x(x - 3)}{5x^2 - 21x - 54}$$. 2. **Recall the domain rule for rational functions:** The domain includes all real numbers except where the denominator is zero because division by zero is undefined. 3. **Set the denominator equal to zero to find excluded values:** $$5x^2 - 21x - 54 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=5$$, $$b=-21$$, and $$c=-54$$. 5. **Calculate the discriminant:** $$\Delta = (-21)^2 - 4 \times 5 \times (-54) = 441 + 1080 = 1521$$ 6. **Find the square root of the discriminant:** $$\sqrt{1521} = 39$$ 7. **Calculate the roots:** $$x = \frac{21 \pm 39}{10}$$ 8. **Find each root:** - $$x = \frac{21 + 39}{10} = \frac{60}{10} = 6$$ - $$x = \frac{21 - 39}{10} = \frac{-18}{10} = -\frac{9}{5}$$ 9. **State the domain:** The function is defined for all real numbers except $$x = 6$$ and $$x = -\frac{9}{5}$$. **Final answer:** $$\boxed{\text{Domain} = \{x \in \mathbb{R} \mid x \neq 6, x \neq -\frac{9}{5} \}}$$