Factorize: 9(x – 2y)² – 4(x – 2y) – 13 by splitting middle term.

For simplicity, let us assume x – 2y = k. So, our expression using this notation is 9k² – 4k – 13.

For factorizing, an expression split the middle term in such a way that the product of the coefficients of the new terms is equal to the product of the coefficients of the first and last terms in the expression.

Here, product of co-effs of first and last terms = 9 × (–13) = –117

So, if the middle term –4k is split into two terms say ak, bk,

then a + b = –4 and ab = –117.

Observe that values –13 and 9 satisfy these equations.

9k² – 4k – 13 = 9k² – 13k + 9k – 13

Observe that k is common for the first two terms and 1 is common for the next two terms.

9k² – 13k + 9k – 13 = k(9k – 13) + 1(9k – 13)

Now, (9k – 13) is the common term.

∴ 9k² – 4k – 13 = (9k – 13)(k + 1)

Thus, the factors of 9k² – 4k – 13 are (9k – 13) and (k + 1).

However, we assumed k = x – 2y

⇒ 9k – 13 = 9(x – 2y) – 13 and k + 1 = x – 2y + 1

Hence, the factors of 9(x – 2y)² – 4(x – 2y) – 13 are (9x – 18y – 13) and (x – 2y + 1).