Simplify completely the quantity x squared plus x minus 12 over quantity x squared minus x minus 20 divided by the quantity 3 x squared minus 24 x plus 45 over quantity 12 x squared minus 48 x minus 60

1. We are asked to simplify the expression : 

2. First thing we can do is to flip the second expression, so make the division a multiplication, and also factorize 3 in the numerator and 12 in the denominator of the second expression as follows:



3. Now factorize each of the quadratric expressions using the following rule:

when we want to factorize , we look for 2 numbers m and n, whose sum is a, and product is b:
for example: in , the 2 numbers we are looking for are clearly -5 and 4, because (-5)+4=-1, (-5)*4=-20, so we write the factorized form (x-5)(x+4) 

Now apply the rule to the whole expression:



4. Simplify equal terms in the numerator and denominator:

we get: