`x gt0`
`{:("Quantity A","Quantity B"),((4x^(2)+2x^(2)+3x+9)/(2x),2x+x+(3)/(2)+(18)/(4x)):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to compare Quantity A and Quantity B given the expressions:

**Quantity A:** \(\frac{4x^2 + 2x^2 + 3x + 9}{2x}\)

**Quantity B:** \(2x + x + \frac{3}{2} + \frac{18}{4x}\)

Let's simplify both quantities step by step.

### Step 1: Simplify Quantity A

We start with Quantity A:

\[ \text{Quantity A} = \frac{4x^2 + 2x^2 + 3x + 9}{2x} \]

Combine like terms in the numerator:

\[ = \frac{(4x^2 + 2x^2) + 3x + 9}{2x} = \frac{6x^2 + 3x + 9}{2x} \]

Now, we can separate the terms in the fraction:

\[ = \frac{6x^2}{2x} + \frac{3x}{2x} + \frac{9}{2x} \]

This simplifies to:

\[ = 3x + \frac{3}{2} + \frac{9}{2x} \]

### Step 2: Simplify Quantity B

Now let's simplify Quantity B:

\[ \text{Quantity B} = 2x + x + \frac{3}{2} + \frac{18}{4x} \]

Combine like terms:

\[ = (2x + x) + \frac{3}{2} + \frac{18}{4x} = 3x + \frac{3}{2} + \frac{18}{4x} \]

Notice that \(\frac{18}{4x}\) can be simplified:

\[ \frac{18}{4x} = \frac{9}{2x} \]

So, we have:

\[ \text{Quantity B} = 3x + \frac{3}{2} + \frac{9}{2x} \]

### Step 3: Compare Quantity A and Quantity B

Now we have:

\[ \text{Quantity A} = 3x + \frac{3}{2} + \frac{9}{2x} \]
\[ \text{Quantity B} = 3x + \frac{3}{2} + \frac{9}{2x} \]

Since both quantities are identical, we conclude:

\[ \text{Quantity A} = \text{Quantity B} \]

### Final Conclusion

Thus, the correct answer is that the two quantities are equal.

**Answer:** The two quantities are equal.

---