To factor the expression provided:

$3(2x - 1)^2(2)(x + 3)^{1/2} + (2x - 1)^{3}(x + 3)^{-1/2}$

We start by identifying the common factors in both terms.

Step 1: Identify the common factors

• Both terms contain $(2x - 1)$.
• The first term has $(2x - 1)^2$, and the second term has $(2x - 1)^3$. The common factor is $(2x - 1)^2$.
• The first term has $(x + 3)^{1/2}$ and the second term has $(x + 3)^{-1/2}$. The common factor here is $(x + 3)^{-1/2}$.

Step 2: Factor out the common factors

Factor out the common factor, which is $(2x - 1)^2(x + 3)^{-1/2}$:

$(2x - 1)^2(x + 3)^{-1/2} \left[ 6(x + 3)^{1} + (2x - 1) \right]$

Step 3: Simplify the expression inside the brackets

Now, simplify the expression inside the brackets:

$6(x + 3) + (2x - 1) = 6x + 18 + 2x - 1 = 8x + 17$

Final factored form:

The expression can be factored as:

$(2x - 1)^2(x + 3)^{-1/2}(8x + 17)$

Would you like more details or further assistance?

Here are 5 related questions to consider:

1. How would you expand the factored expression to verify it's equivalent to the original?
2. What if the expression involved different coefficients, how would the factorization change?
3. Can the expression be solved for specific values of $x$?
4. How would the factoring process differ if the exponents were non-integers?
5. How does factoring help in finding the roots of the equation?

Tip: Factoring is a crucial skill in algebra as it simplifies expressions and helps in solving equations efficiently.