Question

𝑚^2−𝑚−12 / 𝑚+2 ÷ 3𝑚^2−3𝑚−36 / 𝑚^2−9

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify this expression, we need to perform the division first.

Dividing the first fraction by the second fraction is equivalent to multiplying the first fraction by the reciprocal of the second fraction.

Reciprocal of 𝑚+2 is 1/(𝑚+2).
Reciprocal of 3𝑚^2−3𝑚−36 / 𝑚^2−9 is (𝑚^2−9)/(3𝑚^2−3𝑚−36).

So, the expression becomes:

(𝑚^2−𝑚−12 / 𝑚+2) * ((𝑚^2−9)/(3𝑚^2−3𝑚−36))

Now, let's simplify each fraction separately.

(𝑚^2−𝑚−12 / 𝑚+2) can be factored as follows:
(𝑚−4)(𝑚+3) / 𝑚+2

(𝑚^2−9)/(3𝑚^2−3𝑚−36) can also be factored:
(𝑚−3)(𝑚+3) / (3𝑚+12)(𝑚−3)

Now, we can cancel out common factors:

[(𝑚−4)(𝑚+3) / 𝑚+2] * [(𝑚−3)(𝑚+3) / (3𝑚+12)(𝑚−3)]

Since (𝑚−3) cancels out in the numerator and denominator, the simplified expression becomes:

(𝑚−4)(𝑚+3) / 𝑚+2 * 1 / 3𝑚+12

Final result: (𝑚−4)(𝑚+3) / (𝑚+2)(3𝑚+12)

Note: It's always a good practice to check if any of the canceled factors create any excluded values. In this case, 𝑚=3 would lead to division by zero in the original expression.