The problem asks to solve the quadratic equation $x^2 + 14x + 4 = 0$ by completing the square.

1. Problem Description

The problem asks to solve the quadratic equation $x^2 + 14x + 4 = 0$ by completing the square.

2. Solution Steps

We are given the equation $x^2 + 14x + 4 = 0$.

First, isolate the constant term by subtracting 4 from both sides:

$x^2 + 14x = -4$

To complete the square, we need to add $(14/2)^2 = 7^2 = 49$ to both sides:

$x^2 + 14x + 49 = -4 + 49$

$x^2 + 14x + 49 = 45$

Now, the left side is a perfect square:

$(x + 7)^2 = 45$

Take the square root of both sides:

$x + 7 = \pm \sqrt{45}$

$x + 7 = \pm \sqrt{9 \cdot 5}$

$x + 7 = \pm 3\sqrt{5}$

Isolate $x$ by subtracting 7 from both sides:

$x = -7 \pm 3\sqrt{5}$

3. Final Answer

$x = -7 + 3\sqrt{5}, -7 - 3\sqrt{5}$