1. **State the problem:** Solve the quadratic expression $5x^2 + 2x + 45$ for $x$. 2. **Formula used:** To solve a quadratic equation $ax^2 + bx + c = 0$, use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a = 5$, $b = 2$, and $c = 45$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 5 \times 45 = 4 - 900 = -896$$ 5. **Interpret the discriminant:** Since $\Delta < 0$, there are no real roots; the solutions are complex. 6. **Find the complex roots:** $$x = \frac{-2 \pm \sqrt{-896}}{2 \times 5} = \frac{-2 \pm i\sqrt{896}}{10}$$ 7. **Simplify $\sqrt{896}$:** $$896 = 64 \times 14 \Rightarrow \sqrt{896} = 8\sqrt{14}$$ 8. **Final solutions:** $$x = \frac{-2 \pm 8i\sqrt{14}}{10} = \frac{-1}{5} \pm \frac{4i\sqrt{14}}{5}$$ **Answer:** The solutions to $5x^2 + 2x + 45 = 0$ are $$x = -\frac{1}{5} + \frac{4i\sqrt{14}}{5} \quad \text{and} \quad x = -\frac{1}{5} - \frac{4i\sqrt{14}}{5}$$