To simplify the expression \( 27h^8 - 18h^5 + 12h^3h \), we first rewrite \( 12h^3h \) as \( 12h^4 \). So, the expression becomes:
\[ 27h^8 - 18h^5 + 12h^4 \]
Next, we can factor out the greatest common factor (GCF) from each term. The GCF of the coefficients \( 27, -18, \) and \( 12 \) is \( 9 \), and the lowest power of \( h \) is \( h^4 \). Therefore, we factor out \( 9h^4 \):
\[ 27h^8 - 18h^5 + 12h^4 = 9h^4(3h^4 - 2h + 1) \]
To check for equivalent forms among the provided options:
- \( 9h^7 - 6h^4 + 4 \)
- \( 9h^7 - 6h^4 \)
- \( 9h^8 - 6h^5 + 4h \)
- This rational expression does not have any equivalent forms.
The simplest equivalent form we have found through factoring is:
\[ 9h^4(3h^4 - 2h + 1) \]
However, none of the options provided match our factored form or the original expression. Therefore, we conclude that the correct response is:
This rational expression does not have any equivalent forms.