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
