Factoring `x^4 + 27x` makes use of two concepts:

- Factoring a common divisor out of a binomial or greater polynomial
- Sum of two cubes factoring formula

First, factor a common divisor out of the given expression. The common divisor or common factor in the above expression is `x`, and it is factored out as follows:

`x(x^3 + 27)`

Now you have to apply the sum of two cubes formula, because 27 is a cubic number as `27 = 3 * 3 * 3 = 3^3`. So rewrite 27 as `3^3`,

`x(x^3 + 3^3)`

In the above expression, you have a sum of two cubes x^3 and 3^3. Thus it can be factored using the sum of two cubes factoring formula `a^3 + b^3 = (a + b)(a^2 - ab + b^2)`,

`x(x + 3)(x^2 - 3x + 3^2)`

Now simplify by writing `3^2 = 9`,

`x(x + 3)(x^2 - 3x + 9)`

Thus you have completely factored the expression `x^4 + 27x` into `x(x + 3)(x^2 - 3x + 9)`

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