Quiz \( \frac{\text { Factor the expression completely }}{\text { (1) } 3 x^{3 / 2}-9 x^{\frac{1}{2}}+6 x^{-1 / 2}} \) (2) \( (2+x)^{-2 / 3} x+(2+x)^{1 / 3} \)

To tackle the first expression, \( 3 x^{3/2}-9 x^{1/2}+6 x^{-1/2} \), you can start by factoring out the greatest common factor (GCF), which is \( 3 x^{-1/2} \). This simplifies the expression to: \[ 3 x^{-1/2} \left( x^2 - 3x + 2 \right) \] Next, factor the quadratic \( x^2 - 3x + 2 \), which gives you: \[ 3 x^{-1/2} (x - 1)(x - 2) \] So the completely factored form is: \[ 3 (x - 1)(x - 2) x^{-1/2} \] For the second expression, \( (2+x)^{-2/3} x + (2+x)^{1/3} \), you can factor out \( (2+x)^{-2/3} \): \[ (2+x)^{-2/3} \left( x + (2+x) \right) \] This simplifies to: \[ (2+x)^{-2/3} (x + 2 + x) = (2+x)^{-2/3} (2x + 2) \] Further factoring out the 2 gives: \[ 2 (2+x)^{-2/3} (x + 1) \] Thus, the expression is completely factored as: \[ 2 (2+x)^{-2/3} (x + 1) \]