Factor out the greatest common monomial \[ 6 x^{2}+3 x^{3} \]

To factor out the greatest common monomial from the expression \( 6x^{2} + 3x^{3} \), first identify the coefficients and the variable components. The greatest common factor of the coefficients \(6\) and \(3\) is \(3\). For the variable part, both terms share \(x^{2}\) as the lowest power of \(x\). Combining these, we can factor out \(3x^{2}\): \[ 6x^{2} + 3x^{3} = 3x^{2}(2 + x) \] Now the expression is neatly factored! Factoring is like breaking a secret code; it reveals the underlying structure of the expression! Just as detectives piece together clues to solve a mystery, you uncover the common elements in polynomials to simplify and facilitate further operations. If you're looking for practical applications, this skill is crucial not only in algebra but also in fields like engineering and economics, where simplifying expressions can make calculations much easier, allowing us to analyze complex situations or design efficient systems without getting tangled in cumbersome equations!