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

1. Problem Description

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

2. Solution Steps

We start with the equation $x^2 + 14x + 4 = 0$.

We want to complete the square on the left-hand side. To do this, we need to add and subtract $(14/2)^2 = 7^2 = 49$.

$x^2 + 14x + 49 - 49 + 4 = 0$

$(x+7)^2 - 49 + 4 = 0$

$(x+7)^2 - 45 = 0$

$(x+7)^2 = 45$

Taking the square root of both sides, we get

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

Since $45 = 9 \cdot 5$, $\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}$.

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

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

3. Final Answer

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