I. Repeated Difference of Two Squares

The identity $a^2 – b^2 = (a+b)(a-b)$ can sometimes be applied more than once in a single problem.

Note: Difference of Two Squares

Example 1: Factorize $p^4 – 81$

1. Treat as $a^2 – b^2$ where $a = p^2$ and $b = 9$.
   $$\text{Sol: } p^4 – 81 = (p^2)^2 – (9)^2$$
2. Apply the identity:
   $$= (p^2 + 9)(p^2 – 9)$$
3. Factor the second term ($p^2 – 9$) again using the identity, where $a=p$ and $b=3$.
   $$= (p^2 + 9)(p^2 – 3^2)$$
4. Final factorization:
   $$= (p^2 + 9)(p – 3)(p + 3)$$

II. Perfect Square Trinomial (Sum)

The identity $a^2 + 2ab + b^2 = (a+b)^2$ is used for expressions that are the square of a sum.

Note: Perfect Square Sum

Example 1: Factorize $x^2 + 8x + 16$

1. Identify $a^2$ and $b^2$: $x^2$ is $(x)^2$ and $16$ is $(4)^2$. So, $a=x$ and $b=4$.

2. Check the middle term $2ab$: $2(x)(4) = 8x$. This matches.
   $$\text{Sol: } x^2 + 8x + 16 = (x)^2 + 2(x)(4) + (4)^2$$
3. Apply the identity:
   $$= (x + 4)^2$$

Example 2: Factorize $25m^2 + 30m + 9$

1. Identify $a^2$ and $b^2$: $25m^2$ is $(5m)^2$ and $9$ is $(3)^2$. So, $a=5m$ and $b=3$.

2. Check the middle term $2ab$: $2(5m)(3) = 30m$. This matches.
   $$\text{Sol: } 25m^2 + 30m + 9 = (5m)^2 + 2(5m)(3) + (3)^2$$
3. Apply the identity:
   $$= (5m + 3)^2$$

III. Perfect Square Trinomial (Difference)

The identity $a^2 – 2ab + b^2 = (a-b)^2$ is used for expressions that are the square of a difference.

Note: Perfect Square Difference

Example 1: Factorize $4y^2 – 12y + 9$

1. Identify $a^2$ and $b^2$: $4y^2$ is $(2y)^2$ and $9$ is $(3)^2$. So, $a=2y$ and $b=3$.

2. Check the middle term $-2ab$: $-2(2y)(3) = -12y$. This matches.
   $$\text{Sol: } 4y^2 – 12y + 9 = (2y)^2 – 2(2y)(3) + (3)^2$$
3. Apply the identity:
   $$= (2y – 3)^2$$

Example 2: Factorize $a^2 – 2ab + b^2 – c^2$ (Combination)

1. Recognize the Perfect Square: The first three terms form a perfect square: $a^2 – 2ab + b^2 = (a-b)^2$.
   $$\text{Sol: } a^2 – 2ab + b^2 – c^2 = (a-b)^2 – c^2$$
2. Apply Difference of Two Squares: Now the expression is in the form $X^2 – Y^2$, where $X=(a-b)$ and $Y=c$.
   $$= [(a-b) + c][(a-b) – c]$$
3. Final factorization:
   $$= (a – b + c)(a – b – c)$$