Step 1: Understanding the Concept:
The axis of symmetry of a parabola is a straight line that divides the parabola into two identical, mirrored halves.
For a parabola defined by a quadratic equation in \(x\), the axis of symmetry is a vertical line passing through its vertex.
Step 2: Key Formula or Approach:
The approach is to convert the general equation into the standard vertex form by completing the square.
For an equation of the form \(x^2 + Dx + Ey + F = 0\), completing the square yields \((x-h)^2 = 4a(y-k)\).
The vertex is \((h, k)\), and the axis of symmetry is the vertical line \(x = h\).
Step 3: Detailed Explanation:
We start with the given equation:
\[ x^2 + 6x + 4y + 5 = 0 \]
First, isolate the terms containing \(x\) on one side of the equation:
\[ x^2 + 6x = -4y - 5 \]
Next, we complete the square for the quadratic expression in \(x\). We take half of the coefficient of \(x\) (which is \(6/2 = 3\)) and square it (\(3^2 = 9\)).
Add \(9\) to both sides of the equation to maintain balance:
\[ x^2 + 6x + 9 = -4y - 5 + 9 \]
The left side can now be factored as a perfect square:
\[ (x + 3)^2 = -4y + 4 \]
Factor out the coefficient of \(y\) on the right side:
\[ (x + 3)^2 = -4(y - 1) \]
This equation is now in the standard form \((x - h)^2 = 4a(y - k)\).
From this form, we can identify the vertex \((h, k)\) of the parabola as \((-3, 1)\).
Because the squared term involves \(x\), the parabola opens vertically (upwards or downwards).
The axis of symmetry for such a parabola is the vertical line that passes through its vertex.
Therefore, the equation of the axis is \(x = h\), which means \(x = -3\).
Rearranging this to match the given options gives \(x + 3 = 0\).
Step 4: Final Answer:
The axis of the parabola is \(x+3=0\).