To solve the given expression, let's carefully examine the numerator and the denominator.
The given expression is:
[tex]\[
\frac{3^{x^2 + 4} - 2 \cdot 3^{x^2 + 3} + 4 \cdot 3^{x^2 + 1}}{5 \cdot 3^{x^2 + 1} - 2 \cdot 3^{x^2}}
\][/tex]
### Step 1: Simplify the Numerator
First, let's simplify the numerator:
[tex]\[
3^{x^2 + 4} - 2 \cdot 3^{x^2 + 3} + 4 \cdot 3^{x^2 + 1}
\][/tex]
We can factor out \(3^{x^2}\):
[tex]\[
3^{x^2} (3^4) - 2 \cdot 3^{x^2} (3^3) + 4 \cdot 3^{x^2} (3^1)
\][/tex]
This simplifies to:
[tex]\[
3^{x^2} (81) - 2 \cdot 3^{x^2} (27) + 4 \cdot 3^{x^2} (3)
\][/tex]
So the numerator now is:
[tex]\[
3^{x^2} (81 - 54 + 12)
\][/tex]
Combining the constants inside the parentheses:
[tex]\[
3^{x^2} (39)
\][/tex]
Thus, the simplified numerator is:
[tex]\[
39 \cdot 3^{x^2}
\][/tex]
### Step 2: Simplify the Denominator
Next, let's simplify the denominator:
[tex]\[
5 \cdot 3^{x^2 + 1} - 2 \cdot 3^{x^2}
\][/tex]
Again, factor out \(3^{x^2}\):
[tex]\[
5 \cdot 3^{x^2} \cdot 3 - 2 \cdot 3^{x^2}
\][/tex]
Simplifying the terms inside the parentheses:
[tex]\[
5 \cdot 3^{x^2} \cdot 3 = 15 \cdot 3^{x^2}
\][/tex]
So the denominator now is:
[tex]\[
15 \cdot 3^{x^2} - 2 \cdot 3^{x^2}
\][/tex]
Combine the terms:
[tex]\[
(15 - 2) \cdot 3^{x^2}
\][/tex]
So the simplified denominator is:
[tex]\[
13 \cdot 3^{x^2}
\][/tex]
### Step 3: Combine and Simplify
Now we can put the simplified numerator and denominator together:
[tex]\[
\frac{39 \cdot 3^{x^2}}{13 \cdot 3^{x^2}}
\][/tex]
Cancel out \(3^{x^2}\) from the numerator and the denominator:
[tex]\[
\frac{39}{13}
\][/tex]
Simplify the fraction:
[tex]\[
39 \div 13 = 3
\][/tex]
So, the entire expression simplifies to:
[tex]\[
3
\][/tex]
### Step 4: Determine the Correct Option
The question asks for which value of \( x \) the expression simplifies to a particular constant. Given that the simplified expression equals 3, we need to identify the correct option among the values provided.
Since this simplification holds true regardless of the value of \( x \), let’s verify one of the values to ensure correctness. Substituting \( x = 1 \) in the simplified form reaffirms that it holds:
[tex]\[ \frac{3^{1^2 + 4} - 2 \cdot 3^{1^2 + 3} + 4 \cdot 3^{1^2 + 1}}{5 \cdot 3^{1^2 + 1} - 2 \cdot 3^{1^2}} = 3 \][/tex]
The correct answer is:
[tex]\[
\boxed{1}
\][/tex]
