To find the highest common factor (H.C.F.) of the expressions [tex]\((a + 2)^2\)[/tex] and [tex]\(a^2 - 4\)[/tex], follow these steps:
1. Understand the given expressions:
- The first expression is [tex]\((a + 2)^2\)[/tex]. This can be expanded using the binomial theorem, but it's more useful to keep it in its factored form for now.
- The second expression is [tex]\(a^2 - 4\)[/tex]. This is a difference of squares and can be factored as [tex]\((a + 2)(a - 2)\)[/tex].
2. Factor the expressions:
- The first expression, [tex]\((a + 2)^2\)[/tex], is already in its factored form.
- The second expression, [tex]\(a^2 - 4\)[/tex], can be factored as:
[tex]\[
a^2 - 4 = (a + 2)(a - 2)
\][/tex]
3. Identify common factors:
- The factors of [tex]\((a + 2)^2\)[/tex] are [tex]\((a + 2)\)[/tex] repeated twice.
- The factors of [tex]\(a^2 - 4\)[/tex] are [tex]\((a + 2)\)[/tex] and [tex]\((a - 2)\)[/tex].
4. Find the common factors:
- Both expressions have the factor [tex]\((a + 2)\)[/tex] in common.
5. Determine the H.C.F.:
- The common factor between [tex]\((a + 2)^2\)[/tex] and [tex]\(a^2 - 4\)[/tex] is [tex]\((a + 2)\)[/tex].
- Since the H.C.F. is the highest common factor, we take the highest power of [tex]\((a + 2)\)[/tex] common to both expressions. Here, [tex]\((a + 2)\)[/tex] appears in both expressions, but only [tex]\((a + 2)\)[/tex] (to the first power) is common as a factor.
Therefore, the H.C.F. (highest common factor) of the two expressions [tex]\((a + 2)^2\)[/tex] and [tex]\(a^2 - 4\)[/tex] is:
[tex]\[
a + 2
\][/tex]
So the detailed solution concludes that the H.C.F. of [tex]\((a + 2)^2\)[/tex] and [tex]\(a^2 - 4\)[/tex] is [tex]\(a + 2\)[/tex].
