To simplify the expression
[tex]\[
\frac{4}{2 x^2} + \frac{1}{3 x},
\][/tex]
we need to work step-by-step.
### Step 1: Simplify each term individually
First, simplify the term [tex]\(\frac{4}{2 x^2}\)[/tex].
[tex]\[
\frac{4}{2 x^2} = \frac{4}{2 \cdot x^2} = \frac{4}{2} \cdot \frac{1}{x^2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}
\][/tex]
Now, the second term [tex]\(\frac{1}{3 x}\)[/tex] is already simplified.
### Step 2: Combine the terms
Write both terms with the same denominator. The least common denominator (LCD) of [tex]\(x^2\)[/tex] and [tex]\(3x\)[/tex] is [tex]\(3x^2\)[/tex]. Rewrite each fraction so that it has this common denominator.
For [tex]\(\frac{2}{x^2}\)[/tex]:
[tex]\[
\frac{2}{x^2} = \frac{2 \cdot 3}{x^2 \cdot 3} = \frac{6}{3 x^2}
\][/tex]
For [tex]\(\frac{1}{3 x}\)[/tex]:
[tex]\[
\frac{1}{3 x} = \frac{1 \cdot x}{3 x \cdot x} = \frac{x}{3 x^2}
\][/tex]
### Step 3: Add the fractions
Now that both terms have the common denominator [tex]\(3x^2\)[/tex], we can add them together:
[tex]\[
\frac{6}{3 x^2} + \frac{x}{3 x^2} = \frac{6 + x}{3 x^2}
\][/tex]
### Step 4: Simplify the combined expression
The final step is to express this in the form [tex]\(\frac{x + [?]}{x}\)[/tex]. Observe the combined expression:
[tex]\[
\frac{6 + x}{3 x^2} = \frac{x + 6}{3 x^2}
\][/tex]
Separate the expression where the denominator is split:
[tex]\[
\frac{6 + x}{3 x^2} = \frac{x}{3 x^2} + \frac{6}{3 x^2} = \frac{1}{3 x} + \frac{2}{x^2}
\][/tex]
So, putting it into the form we need:
[tex]\[
\frac{x}{x \cdot 3 x} + \frac{6}{x \cdot 3 x^2} = \frac{1}{3 x} + \frac{2}{x^2} = \frac{1 + \frac{6}{x}}{3 x}
\][/tex]
Hence, in the simplified form [tex]\(\frac{x + 6}{3 x^2}\)[/tex], where our numerator [tex]\([?] = 6\)[/tex].
Therefore, the final answer for the simplified form [tex]\(x + [?]\)[/tex] is:
[tex]\[
\boxed{6}
\][/tex]
