x^2–4 ÷ x^2+5x+6

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x^2 + 3x – 10 x^2 + 8x + 15

To simplify the given expression x^2–4 ÷ x^2+5x+6 ÷ x^2 + 3x – 10 ÷ x^2 + 8x + 15, we can follow these steps:

Step 1: Factorize the denominators of each fraction.
The denominators are x^2 + 5x + 6, x^2 + 3x – 10, x^2 + 8x + 15.

To factorize, we need to find two numbers whose sum is equal to the coefficient of x (b) and whose product is equal to the product of the coefficient of x^2 (a) and the constant term (c).

Factors of x^2 + 5x + 6: (x + 2)(x + 3)
Factors of x^2 + 3x – 10: (x + 5)(x – 2)
Factors of x^2 + 8x + 15: (x + 3)(x + 5)

So, the expression becomes:
(x^2 - 4) / [(x + 2)(x + 3)] ÷ [(x + 5)(x - 2)] / [(x + 3)(x + 5)]

Step 2: Simplify the expression by canceling out common factors in the numerator and the denominator.
In this case, we have some common factors that can be canceled out:
- (x + 5) in the numerator and the denominator
- (x + 3) in the numerator and denominator

After canceling out the common factors, the expression becomes:
(x^2 - 4) / [(x + 2)(x - 2)]

Step 3: Further simplify the expression.
The numerator x^2 - 4 is a difference of squares, which can be factored as (x + 2)(x - 2).

So, the final simplified expression is:
[(x + 2)(x - 2)] / [(x + 2)(x + 3)(x - 2)]

The factor (x - 2) in the numerator and the denominator cancels out.

Therefore, the simplified expression is:
(x - 2) / (x + 3)(x + 2)