1. **State the problem:** Factor the expression $3a^4 - 18a^2 b^2 + 3b^4$. 2. **Identify common factors:** Notice each term has a factor of 3, so factor out 3: $$3(a^4 - 6a^2 b^2 + b^4)$$ 3. **Recognize the quadratic form:** The expression inside the parentheses is a quadratic in terms of $a^2$ and $b^2$: $$a^4 - 6a^2 b^2 + b^4 = (a^2)^2 - 2 \cdot 3 \cdot a^2 b^2 + (b^2)^2$$ 4. **Use the perfect square trinomial formula:** $$x^2 - 2xy + y^2 = (x - y)^2$$ Here, $x = a^2$ and $y = b^2$, so: $$a^4 - 6a^2 b^2 + b^4 = (a^2 - b^2)^2$$ 5. **Rewrite the factored form:** $$3(a^2 - b^2)^2$$ 6. **Further factor if possible:** Recall difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ So: $$(a^2 - b^2)^2 = [(a - b)(a + b)]^2 = (a - b)^2 (a + b)^2$$ 7. **Final fully factored form:** $$3 (a - b)^2 (a + b)^2$$