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We are given the following expression to factor completely:

\[
\frac{1}{2} x^{-1/2} (3x + 4)^{1/2} + \frac{3}{2} x^{1/2} (3x + 4)^{-1/2}
\]

Let's factor step by step.

### Step 1: Factor out the common terms

Both terms contain a factor of \(\frac{1}{2}\), so we can factor that out:

\[
\frac{1}{2} \left[ x^{-1/2} (3x + 4)^{1/2} + 3 x^{1/2} (3x + 4)^{-1/2} \right]
\]

### Step 2: Simplify the powers of \(x\) and \((3x + 4)\)

Notice that \(x^{-1/2}\) and \(x^{1/2}\) have different powers of \(x\), as do \((3x + 4)^{1/2}\) and \((3x + 4)^{-1/2}\). Let's look for common factors:

- For \(x\), the lowest power is \(x^{-1/2}\), so we can factor out \(x^{-1/2}\).
- For \((3x + 4)\), the lowest power is \((3x + 4)^{-1/2}\), so we can factor out \((3x + 4)^{-1/2}\).

Factoring these out, we get:

\[
\frac{1}{2} x^{-1/2} (3x + 4)^{-1/2} \left[ (3x + 4) + 3x \right]
\]

### Step 3: Simplify the remaining terms inside the brackets

Inside the brackets, simplify the expression:

\[
(3x + 4) + 3x = 6x + 4
\]

Thus, the expression becomes:

\[
\frac{1}{2} x^{-1/2} (3x + 4)^{-1/2} (6x + 4)
\]

### Step 4: Factor the remaining terms

Notice that \(6x + 4\) can be factored as \(2(3x + 2)\):

\[
\frac{1}{2} x^{-1/2} (3x + 4)^{-1/2} \cdot 2(3x + 2)
\]

The factor of 2 cancels with the \(\frac{1}{2}\), leaving:

\[
x^{-1/2} (3x + 4)^{-1/2} (3x + 2)
\]

### Final Factored Form:

The fully factored form of the given expression is:

\[
x^{-1/2} (3x + 4)^{-1/2} (3x + 2)
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How would the factoring change if different powers of \(x\) were involved?
2. Can this expression be simplified further if \(x\) takes a specific value?
3. What happens to the expression when \(x = 0\)?
4. How would you factor a similar expression involving higher powers of \(x\)?
5. How does factoring help in solving equations with these terms?

### Tip:
Always look for the lowest power of common terms to factor them out efficiently, which simplifies complex expressions quickly.

Factor completely the expression \(\frac{1}{2} x^{-1/2} (3x + 4)^{1/2} + \frac{3}{2} x^{1/2} (3x + 4)^{-1/2}\).

Learn how to factor the complex expression \(\frac{1}{2} x^{-1/2} (3x + 4)^{1/2} + \frac{3}{2} x^{1/2} (3x + 4)^{-1/2}\) step-by-step. This problem covers key algebraic concepts like factoring, exponent rules, and simplifying complex expressions. Suitable for students in Grades 10-12.

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