Formula use: \left(a+b\right)^3=a^3+b^3+3.a.b\left(a+b\right)
Formula use: a^2-b^2=\left(a-b\right)\left(a+b\right)

Explanation

Here, given equation is :
\left(x-y+z\right)^3+\left(x+y-z\right)^3+6x\{x^2-\left(y-z\right)^2\}
Let,
(x-y+z)=a
(x+y-z)=b
If we add a and b then we get :
\left(a+b\right)
\rightarrow\{(x-y+z)+(x+y-z)\}
\rightarrow\left(x-y+z+x+y-z\right)
\therefore 2x _——–(1)
Now, Simplify the equation :
\left(x-y+z\right)^3+\left(x+y-z\right)^3+6x\{x^2-\left(y-z\right)^2\}
Or, \left(x-y+z\right)^3+\left(x+y-z\right)^3+3.2x.\{x-\left(y-z\right)\}.\{x+\left(y-z\right)\} _[a²-b²]
Or, \left(x-y+z\right)^3+\left(x+y-z\right)^3+3.\{(x-y+z)+(x+y-z)\}.\{x+y-z\}.\{x-y+z\} _[a+b=2x]
Now, putting the a & b value, we get :
a^3+b^3+3.\left(a+b\right).ab
\therefore\left(a+b\right)^3
So that, Putting the value :
\left(2x\right)^3 _{[From (1), (a+b)=2x]}
\rightarrow 8x^3
Ans: 8x^3.