How to calculate the expression easily?
I do not know your calculator.
The simplest form I found for your numbers is $$1+(p (11 p (p (13 p (14 p-75)+2100)-2275)+13650)-3003) p^{10}$$ or, if you prefer $$1+p^{10} \Big(2002 p^5-10725 p^4+23100 p^3-25025 p^2+13650 p-3003\Big)$$
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I do not know your calculator.
The simplest form I found for your numbers is $$1+(p (11 p (p (13 p (14 p-75)+2100)-2275)+13650)-3003) p^{10}$$ or, if you prefer $$1+p^{10} \Big(2002 p^5-10725 p^4+23100 p^3-25025 p^2+13650 p-3003\Big)$$