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) #