The problem asks to simplify $H(-t)$, where $H(t) = 6t^2 - 3t + 3$.

To find $H(-t)$, we substitute $-t$ for $t$ in the given expression for $H(t)$:

$H(-t) = 6(-t)^2 - 3(-t) + 3$

Step-by-step simplification:

1. $(-t)^2 = t^2$, so $6(-t)^2 = 6t^2$
2. $-3(-t) = 3t$
3. The constant $3$ remains unchanged.

Thus, $H(-t) = 6t^2 + 3t + 3$.

This is the simplified expression for $H(-t)$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the simplification change if $H(t)$ included higher powers of $t$?
2. What is the significance of even and odd functions in relation to this problem?
3. How would $H(-t)$ change if the linear term had a higher coefficient?
4. Could $H(t)$ represent a real-world situation, and how would changing $t$ to $-t$ affect it?
5. How does substituting $-t$ affect the symmetry of a function?

Tip: Even powers of $t$ remain unchanged when substituting $-t$, while odd powers change sign.