Given that, for all real x, the expression `(x^2+2x+4)/(x^2-2x+4)` lies between `1/3` and 3. The values between which the expression `(9.3^(2x)+6.3^x+4)/(9.3^(2x)-6.3^x+4)` lies are

Options

A. `3^(-1) and 3`
B. `-2 and 0`
C. `-1 and 1`
D. 0 and 2

Explanation

To solve the problem, we start with the given expression:

\[ \frac{x^2 + 2x + 4}{x^2 - 2x + 4} \]

We know that this expression lies between \(\frac{1}{3}\) and \(3\) for all real \(x\). We need to analyze the second expression:

\[ \frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4} \]

### Step 1: Rewrite the second expression

We can rewrite \(9\) as \(3^2\) and \(6\) as \(2 \cdot 3\):

\[ \frac{3^2 \cdot 3^{2x} + 2 \cdot 3 \cdot 3^x + 4}{3^2 \cdot 3^{2x} - 2 \cdot 3 \cdot 3^x + 4} \]

This simplifies to:

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

### Step 2: Let \(y = 3^{x + 1}\)

Now, we can substitute \(y = 3^{x + 1}\):

\[ \frac{y^2 + 2y + 4}{y^2 - 2y + 4} \]

### Step 3: Analyze the expression

We observe that the new expression \(\frac{y^2 + 2y + 4}{y^2 - 2y + 4}\) is similar to the original expression \(\frac{x^2 + 2x + 4}{x^2 - 2x + 4}\). Since we know that the original expression lies between \(\frac{1}{3}\) and \(3\), the same will apply to our new expression.

### Step 4: Conclusion

Thus, we conclude that:

\[ \frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4} \]

also lies between \(\frac{1}{3}\) and \(3\).

### Final Answer

The values between which the expression lies are:

\[ \frac{1}{3} \text{ and } 3 \]