To simplify the rational expression \( \frac{t^2}{6t^2 + 9} \), we start by looking at the denominator. First, we can factor the denominator.
The expression in the denominator is \( 6t^2 + 9 \). Notice that both terms have a common factor of 3. We can factor this out:
\[ 6t^2 + 9 = 3(2t^2 + 3) \]
Now, substituting this back into the original expression gives:
\[ \frac{t^2}{6t^2 + 9} = \frac{t^2}{3(2t^2 + 3)} \]
At this point, we can see there are no common factors to cancel between the numerator and the denominator. Therefore, the rational expression in its simplest form is:
\[ \frac{t^2}{3(2t^2 + 3)} \]
Next, we need to consider the restrictions on the variable \( t \). The denominator cannot be equal to zero, so we set the denominator to zero and solve for \( t \):
\[ 6t^2 + 9 = 0 \]
However, upon simplifying this, we find that:
\[ 6t^2 = -9 \]
\[ t^2 = -\frac{3}{2} \]
This equation does not have any real solutions since the square of a real number cannot be negative. Thus, there are no real values of \( t \) that would make the denominator zero.
In conclusion, the simplified form of the rational expression is:
\[ \frac{t^2}{3(2t^2 + 3)} \]
and the variable \( t \) can be any real number since there are no restrictions.