To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we first want to rewrite the equation in a way that makes it easier to work with.
Step 1: Eliminate the fraction
Multiply the entire equation by 4 to eliminate the fraction:
\[ 4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 0 \]
This simplifies to:
\[ x^2 + 8x + 16 = 0 \]
Step 2: Identify the perfect square trinomial
Now, observe that \( x^2 + 8x + 16 \) can be factored as a perfect square trinomial:
\[ (x + 4)^2 = 0 \]
Step 3: Solve for \( x \)
Set the equation equal to zero:
\[ (x + 4)^2 = 0 \]
Take the square root of both sides:
\[ x + 4 = 0 \]
Now, solve for \( x \):
\[ x = -4 \]
Conclusion
The solution to the equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[ \boxed{-4} \]