Let's simplify the given expression:
[tex]\[
\frac{a^{n+2} - 2a^n + 3a^{n+1}}{a^n + a^{n+1}}
\][/tex]
We will start by rewriting and simplifying the numerator and the denominator.
### Step 1: Rewrite the Terms
- Rewrite [tex]\(a^{n+2}\)[/tex] as [tex]\(a^2 \cdot a^n\)[/tex]
- Rewrite [tex]\(a^{n+1}\)[/tex] as [tex]\(a \cdot a^n\)[/tex]
So, the expression becomes:
[tex]\[
\frac{a^2 \cdot a^n - 2a^n + 3a \cdot a^n}{a^n + a \cdot a^n}
\][/tex]
### Step 2: Factor out [tex]\(a^n\)[/tex]
Now, factor [tex]\(a^n\)[/tex] out of both the numerator and the denominator:
- Numerator: [tex]\(a^n(a^2 - 2 + 3a)\)[/tex]
- Denominator: [tex]\(a^n(1 + a)\)[/tex]
So the expression now is:
[tex]\[
\frac{a^n(a^2 - 2 + 3a)}{a^n(1 + a)}
\][/tex]
### Step 3: Cancel out [tex]\(a^n\)[/tex]
Since [tex]\(a^n\)[/tex] is common to both the numerator and the denominator, we can cancel it out:
[tex]\[
\frac{a^2 - 2 + 3a}{1 + a}
\][/tex]
### Step 4: Simplify the Numerator
The numerator can further be simplified. Rewriting in standard form:
[tex]\[
a^2 + 3a - 2
\][/tex]
So, we have:
[tex]\[
\frac{a^2 + 3a - 2}{1 + a}
\][/tex]
### Summary
By following the above steps, the simplified form of the given expression is:
[tex]\[
\frac{a^2 + 3a - 2}{1 + a}
\][/tex]
This is the final simplified expression.
