1. Problem: Factor the polynomial expression $z^2 + 18zab + 81a^2b^2$ into a perfect square trinomial. 2. Formula: A perfect square trinomial takes the form $$ (x + y)^2 = x^2 + 2xy + y^2 $$ 3. Identify terms: - $x^2 = z^2$ so $x = z$ - $y^2 = 81a^2b^2$ so $y = 9ab$ - Check middle term: $2xy = 2 \times z \times 9ab = 18zab$ which matches the middle term. 4. Therefore, the factorization is: $$ z^2 + 18zab + 81a^2b^2 = (z + 9ab)^2 $$ 5. Explanation: We recognized the polynomial as a perfect square trinomial because the first and last terms are perfect squares and the middle term is twice the product of their square roots. Final answer: $$ (z + 9ab)^2 $$