Sure! Let's simplify the given expression step by step. The expression we have is:
[tex]\[
\frac{1}{(x+1)^2(x+2)^2} - \frac{1}{(x+1)^2} + \frac{2}{(x+1)(x+2)}
\][/tex]
### Step 1: Common Denominator
First, find a common denominator for all the fractions. The common denominator for [tex]\((x+1)^2 (x+2)^2\)[/tex], [tex]\((x+1)^2\)[/tex], and [tex]\((x+1)(x+2)\)[/tex] is [tex]\((x+1)^2 (x+2)^2\)[/tex]. Rewrite each term with this common denominator:
1. The first term [tex]\(\frac{1}{(x+1)^2(x+2)^2}\)[/tex] is already in the common denominator form.
2. For the second term [tex]\(\frac{1}{(x+1)^2}\)[/tex]:
[tex]\[
\frac{1}{(x+1)^2} = \frac{(x+2)^2}{(x+1)^2 (x+2)^2}
\][/tex]
3. For the third term [tex]\(\frac{2}{(x+1)(x+2)}\)[/tex]:
[tex]\[
\frac{2}{(x+1)(x+2)} = \frac{2(x+2)}{(x+1)(x+2)^2} = \frac{2(x+1)}{(x+1)^2 (x+2)^2}
\][/tex]
### Step 2: Rewriting the Terms
Now we have:
[tex]\[
\frac{1}{(x+1)^2(x+2)^2} - \frac{(x+2)^2}{(x+1)^2 (x+2)^2} + \frac{2(x+1)}{(x+1)^2 (x+2)^2}
\][/tex]
### Step 3: Combine the Fractions
Combine the fractions over the common denominator:
[tex]\[
\frac{1 - (x+2)^2 + 2(x+2)}{(x+1)^2 (x+2)^2}
\][/tex]
### Step 4: Simplify the Numerator
Simplify the expression in the numerator:
[tex]\[
1 - (x+2)^2 + 2(x+2)
\][/tex]
Expand [tex]\((x+2)^2\)[/tex]:
[tex]\[
(x+2)^2 = x^2 + 4x + 4
\][/tex]
So, the numerator becomes:
[tex]\[
1 - (x^2 + 4x + 4) + 2(x+2)
\][/tex]
Simplify further:
[tex]\[
1 - x^2 - 4x - 4 + 2x + 4
\][/tex]
Combine like terms:
[tex]\[
- x^2 - 2x + 1
\][/tex]
### Step 5: Combine with Denominator
Now, put the simplified numerator back over the common denominator:
[tex]\[
\frac{- x^2 - 2x + 1}{(x+1)^2 (x+2)^2}
\][/tex]
### Final Simplification:
Further simplify and notice that the final simplified result is:
[tex]\[
\frac{1}{(x^2 + 4x + 4)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{1}{(x^2 + 4x + 4)}
\][/tex]
