Start with dividing the term with the highest exponent in the numerator by the term with the highest exponent in the denominator, so here, divide #a^3# by #a#:
#a^3 / a = a^2#.
So, your first term needs to be #a^2(a+2)#.
#=># Transform the numerator into #a^2(a+2) + 2a^2 + 7a + 6#.
Always pay attention that the value of the numerator doesn't change - here, you have "splitted" the term #4a^2# into the part in #a^2(a+2)# and the rest: #2a^2#.
So far, you've got:
#(a^3 + 4a^2 + 7a + 6) / (a+2)#
# = (a^2(a+2) + 2a^2 + 7a + 6) / (a+2)#
# = (a^2(a+2)) / (a+2) + (2a^2 + 7a + 6) / (a+2)#
# = a^2 + (2a^2 + 7a + 6) / (a+2)#
Your fraction is smaller now. Proceed in the same way:
- divide #2a^2# by #a#, result: #(2a^2) / a = 2a#
- create the expression #2a(a+2)# in the numerator.
- take #4a# (part of your new expression) from #7a#
Now you have:
# a^2 + (2a^2 + 7a + 6) / (a+2)#
#= a^2 + (2a(a+2) + 3a + 6) / (a+2)#
# = a^2 + (2a(a+2)) / (a+2) + (3a + 6) / (a+2)#
# = a^2 + 2a + (3a + 6) / (a+2)#
The last part is easy: #3a# divided by #a# is #3#, and the last expression can be factorized cleanly in #3(a+2) = 3a + 6#.
# a^2 + 2a + (3a + 6) / (a+2)#
# = a^2 + 2a + (3(a+2)) / (a+2)#
# = a^2 + 2a + 3#
This is your final result:
#(a^3 + 4a^2 + 7a + 6) / (a+2) = a^2 + 2a + 3#
