1. **State the problem:** Factorize the expression $$4x^2 - 12y + 9$$. 2. **Identify the type of expression:** This looks like a quadratic expression in terms of $x$ and a constant term involving $y$. 3. **Check if it fits a special product pattern:** The expression resembles a perfect square trinomial of the form $$a^2 - 2ab + b^2 = (a - b)^2$$. 4. **Rewrite the expression:** $$4x^2 - 12y + 9 = (2x)^2 - 2 \times 2x \times 3\sqrt{y} + (3\sqrt{y})^2$$ 5. **Verify the middle term:** $$-2 \times 2x \times 3\sqrt{y} = -12x\sqrt{y}$$, but our middle term is $$-12y$$, not $$-12x\sqrt{y}$$, so this is not a perfect square trinomial. 6. **Try grouping or other factorization methods:** Since the middle term is $$-12y$$ and not involving $x$, the expression cannot be factored as a perfect square. 7. **Check if the expression can be factored as a quadratic in $x$:** Treating $y$ as a constant, the expression is: $$4x^2 + (-12y) + 9$$ Since the middle term does not contain $x$, the expression is not factorable over the integers. 8. **Conclusion:** The expression $$4x^2 - 12y + 9$$ cannot be factorized further using standard algebraic methods. **Final answer:** The expression is already in its simplest form and cannot be factorized further.