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]
