simplify : (p + q + r) whole power 2 + (p - q - r) whole power 2?

Hello,

It bears repeating: you should have applied the well-known following formulas: "Hello", "Please" and "Thanks".
Anyway...

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= = = = = = = = = = = = = = = = = = = = = =
Do you mean
   (𝑝 + 𝑞 + 𝑟)² + (𝑝 – 𝑞 – 𝑟)²
?

In that case, just apply twice the identity:
   (𝛼 + 𝛽 + 𝛾)² = 𝛼² + 𝛽² + 𝛾² + 2𝛼𝛽 + 2𝛽𝛾 + 2𝛼𝛾

Thus:
   (𝑝 + 𝑞 + 𝑟)² + (𝑝 – 𝑞 – 𝑟)²
      = (𝑝 + 𝑞 + 𝑟)² + [𝑝 + (-𝑞) + (-𝑟)]²
      = 𝑝² + 𝑞² + 𝑟² + 2𝑝𝑞 + 2𝑞𝑟 + 2𝑝𝑟 + 𝑝² + (-𝑞)² + (-𝑟)² + 2𝑝(-𝑞) + 2(-𝑞)(-𝑟) + 2𝑝(-𝑟)
      = 𝑝² + 𝑞² + 𝑟² + 2𝑝𝑞 + 2𝑞𝑟 + 2𝑝𝑟 + 𝑝² + 𝑞² + 𝑟² – 2𝑝𝑞 + 2𝑞𝑟 – 2𝑝𝑟
      = 2𝑝² + 2𝑞² + 2𝑟² + 4𝑞𝑟

Regards,
Dragon.Jade :-)
         

What does "whole power" mean.
Perhaps you could write what you want using parentheses, so others could understand.

[ p + q + r ] ² + [ p - q - r ]²

{ [ p + q + r ] - [ p - q - r ] } { [ p + q + r ] + [ p - q - r ] }

[ 2 r ] [ 2 p ] = 4 p r