factorise this quadratic x^2 - 9/x^2 + 5x + 6 in its simplest form

To factorize the expression \( \frac{x^2 - 9}{x^2 + 5x + 6} \), we’ll start by factoring both the numerator and the denominator separately.

### Step 1: Factorize the Numerator
The numerator is \( x^2 - 9 \), which is a difference of squares:
\[
x^2 - 9 = (x + 3)(x - 3)
\]

### Step 2: Factorize the Denominator
The denominator is \( x^2 + 5x + 6 \), which is a simple quadratic expression. We need two numbers that multiply to \( 6 \) and add up to \( 5 \). These numbers are \( 2 \) and \( 3 \):
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]

### Step 3: Rewrite the Expression
Now, substitute these factorizations back into the original expression:
\[
\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x + 3)(x - 3)}{(x + 2)(x + 3)}
\]

### Step 4: Simplify the Expression
We can cancel out \( (x + 3) \) from the numerator and denominator, assuming \( x \neq -3 \) to avoid division by zero:
\[
= \frac{x - 3}{x + 2}
\]

### Final Answer
The simplified form is:
\[
\frac{x - 3}{x + 2}
\]
with the condition that \( x \neq -3 \) and \( x \neq -2 \) to keep the expression defined.

Would you like further details on any step or have additional questions?

Here are some related questions you might explore:
1. How do you factorize the difference of squares in general?
2. Why does cancelling terms require checking for restrictions on \( x \)?
3. How can we verify that \( x - 3 \) over \( x + 2 \) is equivalent to the original expression?
4. What happens if we substitute specific values for \( x \) in the original and simplified expressions?
5. How would you handle this factorization if it was written with addition instead of subtraction in the numerator?

**Tip:** Always check for values that make any denominator zero when simplifying algebraic fractions.

Factorise this quadratic x^2 - 9/x^2 + 5x + 6 in its simplest form.

Factor and simplify the rational expression (x^2 - 9) / (x^2 + 5x + 6) using algebraic techniques. This solution explains factoring quadratics, recognizing a difference of squares, and simplifying rational expressions. Suitable for Grades 9-10.

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