<s>[INST] Is there any easy way to simplify the following term? $[p (p+1) (p+2) (p+3) (p+q)^3] - 4[ p^2 (p+1) (p+2) (p+q)^2 (p+q+3) ]
 +6[ p^3 (p+1) (p+q) (p+q+2) (p+q+3)] -3[ p^4 (p+q+1) (p+q+2) (p+q+3)]$
After the simplification , the result will be:
$3pq \cdot (p^2q + pq^2+2p^2-2pq+2q^2)$
I have tried to simplify it by using the binomial expansion for the power terms and by multiplying , adding, subtracting.   it takes a lot of time.
Is there any alternative easy way to simplify the following term?
 [/INST] Well... This bizzare thing $$[p (p+1) (p+2) (p+3) (p+q)^3] - 4[ p^2 (p+1) (p+2) (p+q)^2 (p+q+3) ]
 +6[ p^3 (p+1) (p+q) (p+q+2) (p+q+3)] -3[ p^4 (p+q+1) (p+q+2) (p+q+3)]$$
Can be split like : $T_1-T_2-3T_2+3T_3+3T_3-3T_4$ , Where $T_i$ denotes the $i^{\text{th}}$ term.
$[p (p+1) (p+2) (p+3) (p+q)^3] - [ p^2 (p+1) (p+2) (p+q)^2 (p+q+3) ] -3[ p^2 (p+1) (p+2) (p+q)^2 (p+q+3) ]
 +3[ p^3 (p+1) (p+q) (p+q+2) (p+q+3)]+ 3[ p^3 (p+1) (p+q) (p+q+2) (p+q+3)] -3[ p^4 (p+q+1) (p+q+2) (p+q+3)]$
Now take the common terms out,
$$p(p+1)(p+2)(p+q)^2[(p+3)(p+q)-p(p+q+3)] \\ 3p^2(p+1)(p+q)(p+q+3)[p(p+q+2)-(p+3)(p+q)] \\ + 3p^3(p+q+2)(p+q+3)[(p+1)(p+q)-p(p+q+1)]$$
Consider only the terms inside the square brackets for now. It's the best to just expand them as it is
$$ (p+3)(p+q)-p(p+q+3) = p^2 +pq +3p+3q-p^2 -pq-3p =3q \\ 
p(p+q+2)-(p+3)(p+q)= p^2+pq+2p -p^2 -pq-2p-2q= -2q \\
(p+1)(p+q)-p(p+q+1)= p^2+pq+p+q-p^2-pq-p =q$$
Now the equation becomes:
$$3pq(p+1)(p+2)(p+q)^2-6p^2q(p+1)(p+q)(p+q+3) +3p^3q(p+q+2)(p+q+3)$$
Again, take the common temrs out, i.e. $3pq$
$$3pq[(p+1)(p+2)(p+q)^2-2p(p+1)(p+q)(p+q+3) +p^2(p+q+2)(p+q+3)$$
Again, split the middle term and common terms out:
$$3pq[(p+1)(p+q)[(p+2)(p+q)-p(p+q+3)]+p^2(p+q+3)[p(p+q+2)-(p+1)(p+q)]]$$
Again, expand the terms in inner square brackets:
$$(p+2)(p+q)-p(p+q+3)=p^2+2p+2q+pq-p^2-pq-3p= 2q-p \\ 
p(p+q+2)-(p+1)(p+q)= p^2+pq+2p-p^2-pq-p-q=p-q
$$
The equation becomes:
$$3pq[(p+1)(p+q)(2q-p)+p^2(p-q)(p+q+3)]$$
I don't think it can be simplified more, so it's best to expand that...
$$=3pq[2p^2q+2pq^2+2pq+2q^2-p^3-p^2q-p^2-pq 
+p^3+p^2q+3p^2-p^2q-pq^2-3pq]$$
Which is indeed equal to
$$3pq[p^2q+pq^2-2pq+2p^2+2q^2]$$
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