Factor 12x^2y-27y-4x^2+9

$12 x^{2} y - 27 y - 4 x^{2} + 9$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(12 x^{2} y - 27 y) - 4 x^{2} + 9$

Factor out the greatest common factor (GCF) from each group.

$3 y (4 x^{2} - 9) - (4 x^{2} - 9)$

Factor the polynomial by factoring out the greatest common factor, $4 x^{2} - 9$ .

$(4 x^{2} - 9) (3 y - 1)$

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$({(2 x)}^{2} - 9) (3 y - 1)$

Rewrite $9$ as $3^{2}$ .

$({(2 x)}^{2} - 3^{2}) (3 y - 1)$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3$ .

$(2 x + 3) (2 x - 3) (3 y - 1)$