What is the numerical coefficient of x^27 in the complete expansion of
(x^2 + 5x + 2)^5 . (x^2 - 7x + 3)^9 ?
here is a hint to get you started. Use the binomial theorem to get
$\displaystyle [(x^2+5x)+2]^5=(x^2+5x)^5+\binom{5}{1}2(x^2+5x)^4+...$
and
$\displaystyle [(x^2-7x)+3]^9=(x^2-7x)^9+\binom{9}{1}3(x^2-7x)^4+...$
Notice the degrees in each.
The first term of the first expansion will have degree 10, the 2nd term will have degree 8.
The first term of the 2nd expansion will have degree 18, then 2nd term will have degree 16.
Why is this observation helpful. Good luck and post back if you get stuck.