Option 3
The solution for given expression is [tex]\frac{(x - 4)(x - 4)}{(x + 3)(x + 1)}[/tex]
Solution:
Given that we have to divide,
[tex]\frac{x^2 -16}{x^2 + 5x + 6} \div \frac{x^2 + 5x + 4}{x^2 -2x - 8}[/tex] ---- (A)
Let us first factorize each term and then solve the sum
Using [tex]a^2 - b^2 = (a + b)(a - b)[/tex]
[tex]x^2 -16 = x^2 - 4^2 = (x + 4)(x -4)[/tex] ----- (1)
[tex]x^2 + 5x + 6 = (x + 2)(x + 3)[/tex] ----- (2)
[tex]x^2 + 5x + 4 = (x+1)(x + 4)[/tex] ---- (3)
[tex]x^2 -2x - 8 = (x-4)(x + 2)[/tex] ---- (4)
Now substituting (1), (2), (3), (4) in (A) we get,
[tex]\frac{(x + 4)(x -4)}{(x +2)(x +3)} \div \frac{(x+1)(x+4)}{(x-4)(x +2)}[/tex]
To do division with fractions, we turn the second fraction upside down and change the division symbol to a multiplication symbol at the same time. Then we treat this as a multiplication problem, by multiplying the numerators and the denominators separately.
[tex]\frac{(x + 4)(x -4)}{(x +2)(x +3)} \times \frac{(x - 4)(x + 2)}{(x + 1)(x + 4)}[/tex]
On cancelling terms we get,
[tex]= \frac{(x -4)(x-4)}{(x + 3)(x + 1)}[/tex]
Thus option 3 is correct
