To solve this problem, we need to find the equation of the line after it is rotated 90 degrees counterclockwise around the origin $(0, 0)$.

Original Equation of the Line

The original equation is:

$2x + y + 3 = 0$

Rotation of a Line by 90 Degrees Counterclockwise

To rotate a line by 90 degrees counterclockwise, we can use the transformation for rotating points:

$(x', y') = (-y, x)$

Substitute $x'$ and $y'$ back into the original line equation. We substitute $x = y'$ and $y = -x'$:

$2(y') + (-x') + 3 = 0$

Simplify this equation:

$2y' - x' + 3 = 0$

Rewriting it in the standard form $Ax + By + C = 0$, we have:

$-x' + 2y' + 3 = 0$

Now, replacing $x'$ back with $x$ and $y'$ back with $y$:

$-x + 2y + 3 = 0$

Conclusion

The equation after rotation is:

$\boxed{x + 2y + 3 = 0}$

So, the correct answer is:

D. x + 2y + 3 = 0

Would you like further clarification or more details?

Here are some additional related questions you might find interesting:

1. What would be the equation of the line if rotated 180 degrees around the origin?
2. How do you rotate a line 90 degrees clockwise around a point other than the origin?
3. How can we determine the intersection points between the original and the rotated lines?
4. What is the geometric interpretation of rotating a line around the origin?
5. How would the equation change if the line were translated instead of rotated?

Tip: When working with rotations, remember that the signs of the coordinates change depending on the direction (clockwise or counterclockwise) and the center of rotation.