Factorise Completely: 12x²y² - 11xy - 1

The expression is $12x^2y^2 - 11xy - 1$. This is a quadratic expression in the form $Au^2 + Bu + C$, where $u = xy$.
Let $u = xy$. The expression becomes $12u^2 - 11u - 1$.

We need to find two numbers that multiply to $A \times C = 12 \times (-1) = -12$ and add up to $B = -11$.
The two numbers are $-12$ and 1, because $(-12) \times 1 = -12$ and $-12 + 1 = -11$.
Rewrite the middle term using these numbers:

Group the first two terms and the last two terms:

Factor out $12u$ from the first group and 1 from the second group:

The common binomial factor is $(u - 1)$:

Replace $u$ with $xy$ in the factored expression: