\(4x^2-12xy=-9y^2\)
\(\Leftrightarrow4x^2-12xy+9y^2=0\)
\(\Leftrightarrow\left(2x\right)^2-2xy+\left(3y\right)^2=0\)
\(\Leftrightarrow\left(2x-3y\right)^2=0\)
\(\Leftrightarrow2x-3y=0\)
\(\Leftrightarrow2x=3y\)
\(\Leftrightarrow\frac{x}{y}=\frac{3}{2}\)