explain to me how to simplify expressions like xsquared+2x-3/x+3

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

### Step 1: Factor the numerator (if possible)
The numerator \( x^2 + 2x - 3 \) is a quadratic expression. To factor it, we need to find two numbers that multiply to give \( -3 \) (the constant term) and add to give \( 2 \) (the coefficient of the middle term, \( x \)).

- The numbers \( 3 \) and \( -1 \) satisfy these conditions:  
  \( 3 \times (-1) = -3 \) and \( 3 + (-1) = 2 \).

So, we can factor the quadratic expression:
\[
x^2 + 2x - 3 = (x - 1)(x + 3)
\]

### Step 2: Simplify the expression
Now, rewrite the entire expression with the factored form of the numerator:
\[
\frac{x^2 + 2x - 3}{x + 3} = \frac{(x - 1)(x + 3)}{x + 3}
\]

Notice that the term \( (x + 3) \) appears in both the numerator and denominator. As long as \( x \neq -3 \) (since that would make the denominator zero, which is undefined), you can cancel out \( (x + 3) \) from both the numerator and denominator.

After cancellation, you're left with:
\[
x - 1
\]

### Final simplified form:
\[
\frac{x^2 + 2x - 3}{x + 3} = x - 1 \quad \text{(for \( x \neq -3 \))}
\]

### Summary:
The expression simplifies to \( x - 1 \), as long as \( x \neq -3 \), because division by zero is not allowed.

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Would you like more details on any part of this process or additional examples?

Here are 5 related questions to deepen your understanding:

1. How do you factor a general quadratic expression like \( ax^2 + bx + c \)?
2. What happens if the denominator in a fraction is zero?
3. Can all quadratic expressions be factored easily? Why or why not?
4. What is the difference between canceling terms and simplifying terms in algebraic expressions?
5. How can you verify if your simplified expression is correct?

**Tip:** Always check for restrictions on the variable (like \( x \neq -3 \)) after simplifying an expression!

Learn how to simplify the quadratic rational expression (x^2 + 2x - 3) / (x + 3) by factoring the numerator and canceling common factors. This guide covers algebraic concepts such as factoring, rational expressions, and simplifying terms, suitable for students in Grades 8-10.

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