1. We are asked to factor the expression: $$32a^2b^2 - 48ab + 18$$. 2. First, identify the greatest common factor (GCF) of all terms. 3. The terms are: $$32a^2b^2$$, $$-48ab$$, and $$18$$. 4. The GCF of the coefficients 32, 48, and 18 is 2. 5. The variable parts: $$a^2b^2$$, $$ab$$, and no variables in 18, so no variable common factor. 6. Factor out 2: $$32a^2b^2 - 48ab + 18 = 2(16a^2b^2 - 24ab + 9)$$ 7. Now factor the quadratic inside the parentheses: $$16a^2b^2 - 24ab + 9$$. 8. Treat $$ab$$ as a single variable, say $$x = ab$$, so the expression becomes: $$16x^2 - 24x + 9$$ 9. Factor $$16x^2 - 24x + 9$$. 10. Find two numbers that multiply to $$16 imes 9 = 144$$ and add to $$-24$$. 11. These numbers are $$-12$$ and $$-12$$. 12. Rewrite the middle term: $$16x^2 - 12x - 12x + 9$$ 13. Group terms: $$(16x^2 - 12x) - (12x - 9)$$ 14. Factor each group: $$4x(4x - 3) - 3(4x - 3)$$ 15. Factor out the common binomial: $$(4x - 3)(4x - 3) = (4x - 3)^2$$ 16. Substitute back $$x = ab$$: $$(4ab - 3)^2$$ 17. Therefore, the fully factored form is: $$2(4ab - 3)^2$$ Final answer: $$\boxed{2(4ab - 3)^2}$$