How can I factor the expression 2x^3 + 54y^3 + 4x + 12y?

To factor the expression 2x³ + 54y³ + 4x + 12y, we can follow these steps:

1. Group the terms: Start by rearranging the terms in pairs for easier factoring.

We can group: (2x³ + 4x) + (54y³ + 12y).

1. Factor out the common terms in each group:

From the first group (2x³ + 4x), we can factor out 2x:

• Factoring gives us: 2x(x² + 2).

From the second group (54y³ + 12y), we can factor out 6y:

• Factoring gives us: 6y(9y² + 2).

Now we rewrite the expression:

2x(x² + 2) + 6y(9y² + 2).

1. Factor out the common terms from the entire expression:

We need to identify if there are any common factors in both parts. In this case, there are no obvious common factors that can be factored out further from the total expression.

Thus, the expression is simplified as follows:

Final Expression: 2x(x² + 2) + 6y(9y² + 2).

If needed, the components (x² + 2) and (9y² + 2) can be checked for further factorization, but they do not factor nicely over the reals.

Therefore, the fully factored form remains:

2x(x² + 2) + 6y(9y² + 2).