1. **State the problem:** Write the equation $$5q^2 - 2q = 4q^2 - 6q + 45$$ in standard form and solve for $q$. 2. **Write the equation in standard form:** Move all terms to one side to set the equation equal to zero. $$5q^2 - 2q - 4q^2 + 6q - 45 = 0$$ 3. **Simplify the equation:** Combine like terms. $$ (5q^2 - 4q^2) + (-2q + 6q) - 45 = 0$$ $$ q^2 + 4q - 45 = 0$$ This is the standard form: $$q^2 + 4q - 45 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula $$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=4$, and $c=-45$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 4^2 - 4 \times 1 \times (-45) = 16 + 180 = 196$$ 6. **Find the square root of the discriminant:** $$\sqrt{196} = 14$$ 7. **Calculate the two solutions:** $$q = \frac{-4 \pm 14}{2 \times 1} = \frac{-4 \pm 14}{2}$$ 8. **First solution:** $$q = \frac{-4 + 14}{2} = \frac{10}{2} = 5$$ 9. **Second solution:** $$q = \frac{-4 - 14}{2} = \frac{-18}{2} = -9$$ **Final answer:** The solutions are $$q = 5$$ and $$q = -9$$.