1. **State the problem:** Simplify the expression $$a^2 y^2 + ay + 2ay^2 + a^2 y^2 + a^2 y$$. 2. **Group like terms:** Group terms with the same variables and powers together: $$a^2 y^2 + a^2 y^2 + ay + 2ay^2 + a^2 y$$ 3. **Combine like terms:** - Combine $$a^2 y^2 + a^2 y^2 = 2a^2 y^2$$ - Combine $$ay$$ and $$a^2 y$$ separately since powers of $$a$$ differ. 4. **Rewrite the expression:** $$2a^2 y^2 + ay + 2ay^2 + a^2 y$$ 5. **Factor where possible:** - Factor $$ay$$ from $$ay + 2ay^2$$: $$ay(1 + 2y)$$ - Factor $$a^2 y$$ from $$2a^2 y^2 + a^2 y$$: $$a^2 y(2y + 1)$$ 6. **Notice that $$1 + 2y$$ and $$2y + 1$$ are the same, so write: $$ay(1 + 2y) + a^2 y(1 + 2y)$$ 7. **Factor out the common factor $$y(1 + 2y)$$:** $$y(1 + 2y)(a + a^2)$$ 8. **Factor $$a$$ from $$a + a^2$$:** $$a(1 + a)$$ 9. **Final simplified expression:** $$a y (1 + 2 y) (1 + a)$$ This is the fully simplified and factored form.