1. Problem: Expand the squares for the expressions: $$(x - 1)^2, (1 - x)^2, (x - 2)^2$$ Step 1: Use the formula $$(a - b)^2 = a^2 - 2ab + b^2$$. $$(x - 1)^2 = x^2 - 2x + 1$$ $$(1 - x)^2 = 1^2 - 2*1*x + x^2 = 1 - 2x + x^2$$ (same as above) $$(x - 2)^2 = x^2 - 4x + 4$$ 2. Problem: Expand the squares for expressions: $$(a + 1)^2, (a - 1)^2, (a + 2)^2$$ Step 1: Use the formula $$(a + b)^2 = a^2 + 2ab + b^2$$. $$(a + 1)^2 = a^2 + 2a + 1$$ $$(a - 1)^2 = a^2 - 2a + 1$$ $$(a + 2)^2 = a^2 + 4a + 4$$ 3. Problem: Expand the squares: $$(2x + 1)^2, (2x - 1)^2, (1 + 2x)^2$$ Step 1: Use $$(a + b)^2 = a^2 + 2ab + b^2$$ and substitute a or b $$(2x + 1)^2 = (2x)^2 + 2*2x*1 + 1^2 = 4x^2 + 4x + 1$$ $$(2x - 1)^2 = (2x)^2 - 2*2x*1 + 1^2 = 4x^2 - 4x + 1$$ $$(1 + 2x)^2$$ is same as $$(2x + 1)^2$$, so $$4x^2 + 4x + 1$$ 4. Problem: Expand squares: $$(x + a)^2, (b - x)^2, (a + c)^2$$ $$(x + a)^2 = x^2 + 2ax + a^2$$ $$(b - x)^2 = b^2 - 2bx + x^2$$ $$(a + c)^2 = a^2 + 2ac + c^2$$ 5. Problem: Expand squares: $$(2x + y)^2, (x + 2y)^2, (x + 5y)^2$$ $$(2x + y)^2 = 4x^2 + 4xy + y^2$$ $$(x + 2y)^2 = x^2 + 4xy + 4y^2$$ $$(x + 5y)^2 = x^2 + 10xy + 25y^2$$ 6. Problem: Expand squares: $$(2m - 3)^2, (3a - 2)^2, (5 - 3b)^2$$ $$(2m - 3)^2 = 4m^2 - 12m + 9$$ $$(3a - 2)^2 = 9a^2 - 12a + 4$$ $$(5 - 3b)^2 = 25 - 30b + 9b^2$$ 7. Problem: Expand squares: $$(5 + 4a)^2, (5 - 4a)^2, (6x - y)^2$$ $$(5 + 4a)^2 = 25 + 40a + 16a^2$$ $$(5 - 4a)^2 = 25 - 40a + 16a^2$$ $$(6x - y)^2 = 36x^2 - 12xy + y^2$$ 8. Problem: Expand squares: $$(2a + 5b)^2, (3m + 2n)^2, (6a + 7c)^2$$ $$(2a + 5b)^2 = 4a^2 + 20ab + 25b^2$$ $$(3m + 2n)^2 = 9m^2 + 12mn + 4n^2$$ $$(6a + 7c)^2 = 36a^2 + 84ac + 49c^2$$ Final answers for 1 to 8 expansions: 1. $x^2 - 2x + 1$, $1 - 2x + x^2$, $x^2 - 4x + 4$ 2. $a^2 + 2a + 1$, $a^2 - 2a + 1$, $a^2 + 4a + 4$ 3. $4x^2 + 4x + 1$, $4x^2 - 4x + 1$, $4x^2 + 4x + 1$ 4. $x^2 + 2ax + a^2$, $b^2 - 2bx + x^2$, $a^2 + 2ac + c^2$ 5. $4x^2 + 4xy + y^2$, $x^2 + 4xy + 4y^2$, $x^2 + 10xy + 25y^2$ 6. $4m^2 - 12m + 9$, $9a^2 - 12a + 4$, $25 - 30b + 9b^2$ 7. $25 + 40a + 16a^2$, $25 - 40a + 16a^2$, $36x^2 - 12xy + y^2$ 8. $4a^2 + 20ab + 25b^2$, $9m^2 + 12mn + 4n^2$, $36a^2 + 84ac + 49c^2$