The problem is to factor
#f^3 + 2f^2 - 64f - 128#
Observe the groupings collected for further simplification:
#color(green)((f^3 + 2f^2)##color(blue)(- 64f - 128)# #color(red)(Expression.1)#
The group #color(green)((f^3 + 2f^2)# can be factored as
#color(green)(f^2(f + 2))# #" "color(red)(Res.1)#
The group #color(blue)(- 64f - 128)# can be factored as
#color(blue)(-64(f+2))# #" "color(red)(Res.2)#
Using our intermediate results #color(red)(Res.1)# and #color(red)(Res.2)# we can write our #color(red)(Expression.1)# as
#color(green)(f^2(f+2)-64(f+2))#
We can now factor them as
#(f^2 - 64)(f+2)# #color(red)(Expression.2)#
Next, we rewrite #(f^2 - 64)# as
#(f^2 - 8^2)#
Using the factoring rule "Difference of Squares" we get
#(f+8)(f - 8)#
Using the above result and our #color(red)(Expression.2)#
we get all of our required factors:
#color(blue)((f+8)(f-8)(f+2))#
