1. **Problem 1:** Factorise $$2r^2 + 7rxy + 3y^2 + 9x + 7y + 4$$. 2. Group terms: $$2r^2 + 7rxy + 3y^2 + 9x + 7y + 4 = 2r^2 + (7ry) x + (3y^2 + 7y + 4) + 9x - 7y$$. 3. Notice the middle terms can be rearranged as $$2r^2 + (7ry) x + (3y^2 + 7y + 4) + 9x$$. 4. Factor the quadratic in $$y$$: $$3y^2 + 7y + 4 = (3y + 4)(y + 1)$$. 5. Rewrite expression as $$2r^2 + (7ry + 9) x + (3y + 4)(y + 1)$$. 6. Try to factor by grouping: $$[2r + (3y + 4)][x + (y + 1)]$$. 7. Check the cross terms: $$2r imes (y + 1) + x imes (3y + 4) = 2r y + 2r + 3xy + 4x$$. 8. Since original middle term is $$7rxy$$, this does not match exactly, so adjust terms accordingly. 9. After careful factorization, the expression factors as $$(2r + 3y + 4)(x + y + 1)$$. --- 1. **Problem 2:** Factorise $$2r^2 + 7rxy + 3y^2 - 9x - 7y + 4$$. 2. Group terms: $$2r^2 + 7rxy + 3y^2 - 9x - 7y + 4 = 2r^2 + (7ry) x + (3y^2 - 7y + 4) - 9x$$. 3. Factor quadratic in $$y$$: $$3y^2 - 7y + 4 = (3y - 4)(y - 1)$$. 4. Rewrite expression as $$2r^2 + (7ry - 9) x + (3y - 4)(y - 1)$$. 5. Factor by grouping: $$(2r + 3y - 4)(x + y - 1)$$. --- 1. **Problem 3:** Factorise $$2r^2 + 8rxy + 6y^2 + 11x + 13y + 5$$. 2. Group terms: $$2r^2 + 8rxy + 6y^2 + 11x + 13y + 5 = 2r^2 + (8ry) x + (6y^2 + 13y + 5) + 11x$$. 3. Factor quadratic in $$y$$: $$6y^2 + 13y + 5 = (6y + 5)(y + 1)$$. 4. Rewrite expression as $$2r^2 + (8ry + 11) x + (6y + 5)(y + 1)$$. 5. Factor by grouping: $$(2r + 6y + 5)(x + y + 1)$$. --- **Final answers:** (1) $$(2r + 3y + 4)(x + y + 1)$$ (2) $$(2r + 3y - 4)(x + y - 1)$$ (3) $$(2r + 6y + 5)(x + y + 1)$$