The general form for factoring a difference of cubes \(a^3 - b^3\) is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
For the expression \(8x^3 - 27\), we can identify \(8x^3\) and \(27\) as cubes:
- \(8x^3 = (2x)^3\)
- \(27 = 3^3\)
Thus, we can set \(a = 2x\) and \(b = 3\).
Now we can substitute \(a\) and \(b\) into the difference of cubes formula:
\[ 8x^3 - 27 = (2x - 3)((2x)^2 + (2x)(3) + (3)^2) \]
Now, simplify \(a^2 + ab + b^2\):
- \((2x)^2 = 4x^2\)
- \((2x)(3) = 6x\)
- \((3)^2 = 9\)
Putting it all together, we have:
\[ 8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9) \]
Thus, the factored form of the expression \(8x^3 - 27\) is:
\[ (2x - 3)(4x^2 + 6x + 9) \]
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