1. **State the problem:** Simplify the expression $$\frac{3(t + h)^2 - 3}{h}$$ and find its simplified form. 2. **Recall the formula and rules:** We will expand the square, simplify the numerator, and then divide by $h$. Remember that $h \neq 0$ to avoid division by zero. 3. **Expand the square:** $$ (t + h)^2 = t^2 + 2th + h^2 $$ 4. **Substitute back:** $$ 3(t + h)^2 - 3 = 3(t^2 + 2th + h^2) - 3 = 3t^2 + 6th + 3h^2 - 3 $$ 5. **Rewrite the expression:** $$ \frac{3t^2 + 6th + 3h^2 - 3}{h} $$ 6. **Factor the numerator where possible:** $$ 3t^2 - 3 + 6th + 3h^2 = 3(t^2 - 1) + 6th + 3h^2 $$ 7. **Group terms to factor $h$ out:** $$ \frac{3(t^2 - 1) + 6th + 3h^2}{h} = \frac{3(t^2 - 1)}{h} + \frac{6th}{h} + \frac{3h^2}{h} $$ 8. **Simplify terms with $h$ in denominator:** $$ = \frac{3(t^2 - 1)}{h} + 6t + 3h $$ 9. **Note:** The term $\frac{3(t^2 - 1)}{h}$ cannot be simplified further without additional context (e.g., limit as $h \to 0$). **Final simplified expression:** $$ \frac{3(t + h)^2 - 3}{h} = \frac{3(t^2 - 1)}{h} + 6t + 3h $$