This section covers two key methods used when simple H.C.F. extraction isn’t enough: Factoring by Grouping and using the Difference of Two Squares identity.

I. Factoring by H.C.F. (Review and Advanced Example)

This is the continuation of taking out the Highest Common Factor (H.C.F.).

Note: Simple H.C.F. (Subtraction)

Example 1: Factorize $22y – 33z$

1. Find the H.C.F. of 22 and 33: The H.C.F. is $11$.

2. Factor it out:
   $$\text{Sol: } 22y – 33z = 11 \times 2 \times y – 11 \times 3 \times z$$
   $$= 11(2y – 3z)$$

Example 2: Factorize $10x^2 – 18x^3 + 14x^4$

1. Find the H.C.F. of Coefficients (10, 18, 14): The H.C.F. is $2$.

2. Find the H.C.F. of Variables ($x^2, x^3, x^4$): The lowest power is $x^2$.

3. Overall H.C.F. is $2x^2$.
   $$\text{Sol: } 10x^2 – 18x^3 + 14x^4 = 2x^2(5 – 9x + 7x^2)$$

II. Factoring by Grouping

This technique is used when an expression has four terms. You group the terms into pairs, find the H.C.F. for each pair, and then factor out the common binomial.

Note: Factoring by Grouping

Example 1: Factorize $2xy + 2y + 3x + 3$

1. Group the terms: $(2xy + 2y) + (3x + 3)$

2. Factor H.C.F. from each group:
   $$\text{Sol: } 2y(x + 1) + 3(x + 1)$$
3. Factor out the common binomial $(x+1)$:
   $$= (2y + 3)(x + 1)$$

Example 2: Factorize $6xy – 4y + 6 – 9x$

1. Rearrange the terms (if necessary) and group: $(6xy – 4y) + (6 – 9x)$

2. Factor H.C.F. from each group:

   • $6xy – 4y = 2y(3x – 2)$

   • $6 – 9x = 3(2 – 3x)$

3. To get a common binomial, factor out $-1$ from the second group:
   $$3(2 – 3x) = -3(3x – 2)$$
4. Rewrite the expression and factor out the common binomial $(3x-2)$:
   $$= 2y(3x – 2) – 3(3x – 2)$$
   $$= (2y – 3)(3x – 2)$$

III. Difference of Two Squares

This is one of the most frequently used identities for factorization.

Note: Difference of Two Squares

Example 1: Factorize $49y^2 – 36$

1. Express each term as a perfect square ($a^2$ and $b^2$):
   $$\text{Sol: } 49y^2 – 36 = (7y)^2 – (6)^2$$
2. Apply the identity $(a+b)(a-b)$ where $a=7y$ and $b=6$:
   $$= (7y + 6)(7y – 6)$$