<s>[INST] How do I factor trinomials of the format $ax^2 + bxy + cy^2$? Take, for example, the polynomial $15x^2 + 5xy - 12y^2$.
 [/INST] I will assume that the coefficients $a$, $b$, and $c$ are real, with $a \ne 0$, and we are factoring over the reals.
We can write our polynomial as 
$$\frac{1}{4a}\left(4a^2x^2+4abxy+4ac\right).$$
Complete the square. We get
$$\frac{1}{4a}\left((2ax+by)^2-(b^2-4ac)y^2\right).$$
If we are trying to factor over the reals, and $b^2-4ac$ is negative, there will be no factorization. But if $b^2-4ac \ge 0$, then we can factor as 
$$\frac{1}{4a}\left(\left(2ax+\left(b-\sqrt{b^2-4ac}\right)y\right)\left(2ax+\left(b+\sqrt{b^2-4ac}\right)y\right)\right).$$
If we wish, we can absorb $\frac{1}{4a}$ in various ways into the "main" factors.
For your particular example, the factorization is 
$$\frac{1}{60}\left(30x+(5-\sqrt{745})y\right)\left(30x+(5+\sqrt{745})y\right).$$
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