1. We are asked to simplify the expression : [tex] \frac{ x^{2} +x-12}{ x^{2} -x-20}/ \frac{ 3x^{2} -24x+45}{ 12x^{2} -48x-60}[/tex]
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:
[tex]\frac{ x^{2} +x-12}{ x^{2} -x-20}* \frac{12( x^{2} -4x-5)}{3(x^{2} -8x+15)}[/tex]
3. Now factorize each of the quadratric expressions using the following rule:
when we want to factorize [tex] x^{2} +ax+b[/tex], we look for 2 numbers m and n, whose sum is a, and product is b:
for example: in [tex]x^{2} -x-20[/tex], 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:
[tex]\frac{(x-3)(x+4)}{(x-5)(x+4)}* \frac{4(x-5)(x+1)}{(x-5)(x-3)}[/tex]
4. Simplify equal terms in the numerator and denominator:
we get: [tex] \frac{4(x+1)}{(x-5)} [/tex]
Answer: [tex] \frac{4(x+1)}{(x-5)} [/tex]
