1. State the problem: Factor the expression $4x^2 + 36xw + 81w^2$. 2. Notice it is a perfect-square trinomial of the form $a^2 + 2ab + b^2$. 3. Identify $a=2x$ and $b=9w$ because then $a^2=(2x)^2=4x^2$ and $b^2=(9w)^2=81w^2$. 4. Check the middle term using $2ab$: 5. $$2ab=2(2x)(9w)=4\cdot 9\,xw=36xw$$ 6. Since all three terms match, write the factorization: 7. $$4x^2 + 36xw + 81w^2=(2x)^2+2(2x)(9w)+(9w)^2=(2x+9w)^2$$ 8. Final answer: $\,(2x+9w)^2$.\n