Factors of \( 8 x^{3}+27 y^{3} \) are:

To factor \( 8x^{3} + 27y^{3} \), we can recognize that this expression is a sum of cubes. It can be rewritten as: \[ (2x)^{3} + (3y)^{3} \] Using the formula for factoring a sum of cubes, \( a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) \), where \( a = 2x \) and \( b = 3y \), we can proceed as follows: 1. \( a + b = 2x + 3y \) 2. \( a^{2} = (2x)^{2} = 4x^{2} \) 3. \( ab = (2x)(3y) = 6xy \) 4. \( b^{2} = (3y)^{2} = 9y^{2} \) Now we can plug these into the formula: \[ 8x^{3} + 27y^{3} = (2x + 3y)(4x^{2} - 6xy + 9y^{2}) \] So the factors are \( (2x + 3y)(4x^{2} - 6xy + 9y^{2}) \). This is a beautiful example of how algebra can unveil hidden patterns, turning a complicated-looking expression into a more manageable product. Plus, it's always satisfying to break things down into simpler parts!