Transform the Fraction Equation: Simplifying 1 + y/x + x/4y = 0

To solve this problem, let's work through these steps:

• Step 1: Start with the given equation: $1+\frac{y}{x}+\frac{x}{4y}=0$.
• Step 2: Clear the fractions by multiplying all terms by $4xy$, the common denominator.
• Step 3: This gives us: $4xy + 4y^2 + x^2 = 0$.
• Step 4: Rearrange the terms for clarity: $x^2 + 4xy + 4y^2 = 0$.
• Step 5: Recognize this as the perfect square: $(x + 2y)^2 = 0$.

This simplification results in the equation: $(x+2y)^2 = 0$.

Therefore, the solution to the problem is $(x+2y)^2 = 0$.