How can the algebraic expression left parenthesis x plus 1 right parenthesis Superscript one half Baseline minus one third left parenthesis x plus 1 right parenthesis Superscript three halves be facto?

To factor the expression ((x + 1)^{1/2} - \frac{1}{3}(x + 1)^{3/2}), we can first factor out the common term ((x + 1)^{1/2}). This gives us ((x + 1)^{1/2} \left(1 - \frac{1}{3}(x + 1)\right)). Simplifying the term inside the parentheses results in ((x + 1)^{1/2} \left(1 - \frac{1}{3}x - \frac{1}{3}\right)), which can be further simplified to ((x + 1)^{1/2} \left(-\frac{1}{3}x + \frac{2}{3}\right)).