Let's Factorise the #color(red)(NUMERATOR)# first.
The Numerator is #color(red)(4g^2 - 64g + 252)#
# = 4(g^2 - 16g + 63)# (4 was the factor common to all terms)
Now we need to factorise #color(blue) (g^2 - 16g + 63) #
We can use Splitting the Middle Term technique to factorise this.
It is in the form #ax^2 + bx + c# where #a=1, b=-16, c= 63#
To split the middle term, we need to think of two numbers #N_1 and N_2# such that:
#N_1*N_2 = a*c and N_1+N_2 = b#
#N_1*N_2 = (1)*(63) and N_1+N_2 = 16#
#N_1*N_2 = 63 and N_1+N_2 = -16#
After Trial and Error, we get #N_1 = -7 and N_2 = -9#
#(-7)*(-9) = 63# and #(-7) + (-9) = -16#
So we can write the expression in blue as
#color(blue) (g^2 - 7g -9g+ 63) #
#=g(g-7)-9(g-7)#
#= (g-7)*(g-9)#
The Numerator can be written as #color(red)(4(g-7)*(g-9))#
The expression we have been given is
#(4g^2 - 64g + 252)/(g-7)#
After the numerator was factorised, the Expression can now be written as :
#(4(g-7)*(g-9))/(g-7)#
# =(4*cancel((g-7))*(g-9))/cancel((g-7))#
# = 4*(g-9) #
#(4g^2 - 64g + 252)/(g-7)# = # 4*(g-9) #
