Decide whether $x^2+y^2+2x+4y+10=0$ represents a real circle. Explain your reasoning to justify.

Yes, it represents a circle with center $(-1,-2)$ and imaginary radius $\sqrt{5}$, which is acceptable

No, completing the square gives $(x+1{)}^2+(y+2{)}^2=-5$, so $R^2$ is negative, and no real circle exists

Yes, the presence of $x^2$ and $y^2$ terms guarantees a circle, regardless of the constant term

No, the equation represents a parabola