I assume your question is (r + 6)/(r^2 - r - 6)r+6r2−r−6
Let's Factorise the color(red)(DENOMINATOR)DENOMINATOR first.
The Denominator is color(red)(r^2 - r - 6)r2−r−6
We can use Splitting the Middle Term technique to factorise this.
It is in the form ax^2 + bx + cax2+bx+c where a=1, b=-1, c= -6a=1,b=−1,c=−6
To split the middle term, we need to think of two numbers N_1 and N_2N1andN2 such that:
N_1*N_2 = a*c and N_1+N_2 = bN1⋅N2=a⋅candN1+N2=b
N_1*N_2 = (1)*(-6) and N_1+N_2 = -1N1⋅N2=(1)⋅(−6)andN1+N2=−1
N_1*N_2 = -6 and N_1+N_2 = -1N1⋅N2=−6andN1+N2=−1
After Trial and Error, we get N_1 = 2 and N_2 = -3N1=2andN2=−3
(2)*(-3) = -6(2)⋅(−3)=−6 and (2) + (-3) = -1(2)+(−3)=−1
So we can write the denominator as
color(red)(r^2 +2r -3r - 6)r2+2r−3r−6
= r*(r+2) - 3*(r+2)=r⋅(r+2)−3⋅(r+2)
= (r+2)*(r-3)=(r+2)⋅(r−3)
The Denominator can be written as color(red)((r-2)*(r+3))(r−2)⋅(r+3)
The expression we have been given is
(r + 6)/(r^2 - r - 6)r+6r2−r−6
After the denominator was factorised, the Expression can now be written as :
((r+6))/((r+2)*(r-3))(r+6)(r+2)⋅(r−3)
(r + 6)/(r^2 - r - 6)r+6r2−r−6 = ((r+6))/((r+2)*(r-3))(r+6)(r+2)⋅(r−3)
