To simplify the expression \(\frac{x+3}{x^2 + 2x + 1} - \frac{x-1}{x^2 + x - 2}\), follow these steps:

### Step 1: Factor the Denominators
1. \(x^2 + 2x + 1\) can be factored as \((x+1)^2\).
2. \(x^2 + x - 2\) can be factored as \((x+2)(x-1)\).

So the expression becomes:
\[
\frac{x+3}{(x+1)^2} - \frac{x-1}{(x+2)(x-1)}
\]

### Step 2: Simplify the Expression
Notice that in the second fraction, the \(x-1\) terms can cancel out:
\[
\frac{x+3}{(x+1)^2} - \frac{1}{x+2}
\]

### Step 3: Find a Common Denominator
The common denominator between \((x+1)^2\) and \(x+2\) is \((x+1)^2(x+2)\).

Rewrite each fraction with the common denominator:
\[
\frac{(x+3)(x+2)}{(x+1)^2(x+2)} - \frac{(x+1)^2}{(x+1)^2(x+2)}
\]

### Step 4: Combine the Fractions
Combine the numerators:
\[
\frac{(x+3)(x+2) - (x+1)^2}{(x+1)^2(x+2)}
\]

### Step 5: Expand and Simplify the Numerator
Expand both terms in the numerator:
1. \((x+3)(x+2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6\)
2. \((x+1)^2 = x^2 + 2x + 1\)

Subtract these to get:
\[
(x^2 + 5x + 6) - (x^2 + 2x + 1) = 3x + 5
\]

So the simplified expression is:
\[
\frac{3x + 5}{(x+1)^2(x+2)}
\]

This is the simplified form of the given expression.

Would you like any further details or have any questions?

### Related Questions:
1. How do you factor quadratic expressions?
2. What are the steps to find a common denominator in rational expressions?
3. How can you simplify complex fractions?
4. What are the conditions under which a fraction can be simplified by canceling terms?
5. How does expanding polynomials help in simplifying expressions?

### Tip:
Always factor the denominators first when dealing with rational expressions; it simplifies the process of finding common denominators and combining fractions.

Learn how to simplify the expression (x+3)/(x^2 + 2x + 1) - (x-1)/(x^2 + x - 2) using factoring techniques and common denominators. This solution covers key algebraic concepts such as factoring quadratics and simplifying rational expressions. Suitable for students in Grades 9-11.

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