Formula use:

\left(a-b\right)^3=a^3-3a^2b+3ab^2-b^3
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: 27x^3-{27x}^2+9x-65
Factorize
27x^3-{27x}^2+9x-65
\rightarrow27x^3-{27x}^2+9x-1-64
\rightarrow\left(3x\right)^3-3.\left(3x\right)^2.1+3.3x.\left(1\right)^2-\left(1\right)^3-64
\rightarrow\left(3x-1\right)^3-\left(4\right)^3
\rightarrow\{\left(3x-1\right)-4\}\{\left(3x-1\right)^2+\left(3x-1\right).4+\left(4\right)^2\}
\rightarrow\left(3x-5\right)\left(9x^2-2.3x.1+1+12x-4+16\right)
\therefore\left(3x-5\right)\left({9x}^2+6x+13\right)
Ans: \left(3x-5\right)\left({9x}^2+6x+13\right)