Transforming the equation into a quadratic equation, we have:
[tex]\begin{gathered} 2x+6=\frac{-5}{x} \\ 2x^2+6x=-5\text{ (Multiplying on both sides by x)} \\ 2x^2+6x+5=0\text{ (Adding 5 to both sides of the equation)} \\ \text{ Using the quadratic equation with a=2, b=6, c=5},\text{ we have:} \\ \frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \frac{-(6)\pm\sqrt[]{(6)^2-4(2)(5)}}{2(2)}\text{ (Replacing the values)} \\ \frac{-6\pm\sqrt[]{36^{}-40}}{4}\text{ (Raising 6 to the power of 2)} \\ \frac{-6\pm\sqrt[]{-4}}{4}\text{ (Subtracting)} \\ \frac{-6\pm(\sqrt[]{4})(\sqrt[]{-1})}{4}\text{ (Rewriting the expression)} \\ \frac{-6\pm2i}{4}\text{ (Taking the square root of 4)} \\ \text{First answer:} \\ \frac{-6+2i}{4}=\frac{2(-3+i)}{4}=\frac{-3+i}{2}\text{ (Simplifying)} \\ \text{ Second answer:} \\ \frac{-6-2i}{4}=\frac{-2(3+i)}{4}=\frac{-3-i}{2}\text{ (Simplifying)} \\ \text{The correct option is the option }B \end{gathered}[/tex]
