Ta có:
$15(3x^2+5xy-4y^2)-38(5x^2-9xy-3y^2)=0$
$\Leftrightarrow -145x^2+417xy+54y^2=0$
$\Leftrightarrow (x-3y)(145x+18y)=0$
$\Leftrightarrow \left[\begin{array}{l}x=3y\\x=\dfrac{-18}{145}y\end{array}\right.$
Với $x=3y$ suy ra: $38y^2=38 \Leftrightarrow y=\pm1$
Với $x=\dfrac{-18}{145}y$ $\Rightarrow \dfrac{-96178}{21025}y^2=38$, vô nghiệm.
Suy ra: $(x;y) = (1;3);(-1;-3)$