To factorize the expression [tex]\(a^2 + 2ab + b^2 - 4\)[/tex], let's proceed step-by-step.
1. Identify and group terms:
The given expression is:
[tex]\[
a^2 + 2ab + b^2 - 4
\][/tex]
We notice the first three terms form a perfect square trinomial:
[tex]\[
a^2 + 2ab + b^2
\][/tex]
2. Factor the perfect square trinomial:
The trinomial [tex]\(a^2 + 2ab + b^2\)[/tex] can be factored as:
[tex]\[
(a + b)^2
\][/tex]
Hence, the expression becomes:
[tex]\[
(a + b)^2 - 4
\][/tex]
3. Recognize the difference of squares:
The expression [tex]\((a + b)^2 - 4\)[/tex] is a difference of squares, where:
[tex]\[
(a + b)^2 - 4 = (a + b)^2 - 2^2
\][/tex]
4. Apply the difference of squares formula:
The difference of squares formula is [tex]\(x^2 - y^2 = (x - y)(x + y)\)[/tex]. Here, we can set [tex]\(x = (a + b)\)[/tex] and [tex]\(y = 2\)[/tex], yielding:
[tex]\[
((a + b) - 2)((a + b) + 2)
\][/tex]
5. Final factorized form:
So, the completely factorized form of [tex]\(a^2 + 2ab + b^2 - 4\)[/tex] is:
[tex]\[
(a + b - 2)(a + b + 2)
\][/tex]
Therefore, the expression [tex]\(a^2 + 2ab + b^2 - 4\)[/tex] factorizes completely to [tex]\((a + b - 2)(a + b + 2)\)[/tex].
