Solve for \( x \).

[tex]\[ 3x = 6x - 2 \][/tex]



Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
[tex]$\begin{array}{l}h(x)=\frac{3 x}{x^2+5 x+12} \\ h\left(x^2\right)=\square\end{array}$[/tex]
-----

Response:
Given the function \( h(x) = \frac{3x}{x^2 + 5x + 12} \),

find [tex]\( h(x^2) \)[/tex].



Answer :

To find the expression \( h(x^2) \) given the function \( h(x) = \frac{3x}{x^2 + 5x + 12} \), we need to evaluate \( h \) at \( x^2 \) instead of \( x \). Here's a detailed, step-by-step solution:

1. Start with the given function \( h(x) \):
[tex]\[ h(x) = \frac{3x}{x^2 + 5x + 12} \][/tex]

2. To find \( h(x^2) \), we replace \( x \) with \( x^2 \) in the definition of \( h(x) \):
[tex]\[ h(x^2) = \frac{3(x^2)}{(x^2)^2 + 5(x^2) + 12} \][/tex]

3. Simplify the expression inside the function:
- The numerator becomes \( 3(x^2) \) or \( 3x^2 \).
- In the denominator, \((x^2)^2\) becomes \( x^4 \) and \( 5(x^2) \) is \( 5x^2 \).
Thus, the entire denominator simplifies to \( x^4 + 5x^2 + 12 \).

4. Putting it all together, we get:
[tex]\[ h(x^2) = \frac{3x^2}{x^4 + 5x^2 + 12} \][/tex]

Thus, the simplified form of \( h(x^2) \) is:
[tex]\[ h(x^2) = \frac{3x^2}{x^4 + 5x^2 + 12} \][/tex]

So the completed expression is:
[tex]\[ h(x^2) = \frac{3x^2}{x^4 + 5x^2 + 12} \][/tex]