Formula use:

a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)
\left(a+b\right)^2=a^2+2ab+b^2

Explanation

Given expression is: 8\left(x+y\right)^3-{27\left(y+z\right)}^3
Factorize
8\left(x+y\right)^3-{27\left(y+z\right)}^3
\rightarrow\{2\left(x+y\right)\}^3-\{3\left(y+z\right)\}^3
\rightarrow\left(2x+2y\right)^3-\left(3y+3z\right)^3
\rightarrow\left(2x+2y-3y-3z\right)\{\left(2x+2y\right)^2+\left(2x+2y\right).\left(3y+3z\right)+\left(3y+3z\right)^2\}
\rightarrow\left(2x-y-3z\right)\left({4x}^2+8xy+{4y}^2+6xy+{6y}^2+6xz+6yz+9y^2+18yz+9z^2\right)
\therefore\left(2x-y-3z\right)\left({4x}^2+14xy+19y^2+6xz+24yz+9z^2\right)
Ans: \left(2x-y-3z\right)\left({4x}^2+14xy+19y^2+6xz+24yz+9z^2\right)