Factorize \[2x^3 + 54y^3 - 4x - 12y\]

Hint: The given equation is \[2{x^3} + 54{y^3} - 4x - 12y\] so we can solve or factorize this polynomial by rearranging it and taking the common factors after that we can use suitable algebraic identity. And further we can take common factors if any term is present.

Complete step by step answer:
Now let us consider the given expression \[2{x^3} + 54{y^3} - 4x - 12y\]
To factorize the above expression let us take 2 as a common factor as all the terms in the given expression are multiple of 2. Then the above equation become
\[2({x^3} + 27{y^3} - 2x - 6y)\]
The above expression can be written as
 \[2({x^3} + 27{y^3}) - 2(2x + 6y)\]
Again taking 2 as a common factor from the second term we get
\[2({x^3} + 27{y^3}) - 4(x + 3y)\]
Now we can note that the first term is the form of \[{a^3} + {b^3}\] therefore using the identity \[{a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})\] the above expression can be written as
\[ \Rightarrow 2(x + 3y)({x^2} - 3xy + 9{y^2}) - 4(x + 3y)\]
We can note that \[2(x + 3y)\]is present in both the terms so taking it as a common term we get
\[ \Rightarrow 2(x + 3y)\left[ {({x^2} - 3xy + 9{y^2}) - 2} \right]\]
\[ \Rightarrow 2(x + 3y)\left[ {{x^2} - 3xy + 9{y^2} - 2} \right]\]
Further we cannot simplify the above expression so the required factors of the given polynomial is
\[2(x + 3y)\left[ {({x^2} - 3xy + 9{y^2}) - 2} \right]\]

Note: In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems.
The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In mathematics there are 8 mathematical identities.This algebraic identity is equality which is true for all the values of the variables.