9х^2 - 24ху + 7y^2

The expression \( 9x^2 - 24xy + 7y^2 \) is a quadratic expression in terms of the variables \( x \) and \( y \).

To factor this expression, we can look for two numbers that multiply to give the product of the coefficient of \( x^2 \) (which is 9) and the coefficient of \( y^2 \) (which is 7) and that add up to the coefficient of \( xy \) (which is -24).

1. The product of the coefficients is \( 9 \times 7 = 63 \).
2. We need two numbers that multiply to 63 and add to -24. These numbers are -21 and -3.

Now we can rewrite the middle term:

\[ 9x^2 - 21xy - 3xy + 7y^2 \]

Next, we can factor by grouping:

\[ = (9x^2 - 21xy) + (-3xy + 7y^2) \]

Factoring out the common factors in each group:

\[ = 3x(3x - 7y) - y(3x - 7y) \]

Now we factor out the common binomial factor:

\[ = (3x - 7y)(3x - y) \]

Thus, the factored form of the expression \( 9x^2 - 24xy + 7y^2 \) is:

\[ (3x - 7y)(3x - y) \]