1. **Problem 1: Add the polynomials** $(3x^2 - 5y^2 + xy) + (8y^2 - 4xy - 7x^2)$. 2. **Step 1: Write the expression**: $$ (3x^2 - 5y^2 + xy) + (8y^2 - 4xy - 7x^2) $$ 3. **Step 2: Group like terms**: $$ (3x^2 - 7x^2) + (-5y^2 + 8y^2) + (xy - 4xy) $$ 4. **Step 3: Simplify each group**: $$ 3x^2 - 7x^2 = \cancel{3x^2} - 7x^2 = -4x^2 $$ $$ -5y^2 + 8y^2 = \cancel{-5y^2} + 8y^2 = 3y^2 $$ $$ xy - 4xy = \cancel{xy} - 4xy = -3xy $$ 5. **Step 4: Write the simplified polynomial**: $$ -4x^2 + 3y^2 - 3xy $$ --- 6. **Problem 2: Identify equivalent polynomials among:** (i) $3x^2 + 3x - 4 + 2x^2 - 6x - 3$ (ii) $x^2 + 12 + 2x - 5 - 5x + 4x^2$ (iii) $3x^2 - 6x + 2x^2 + 3 + 3x - 10$ 7. **Step 1: Simplify each polynomial by combining like terms**: (i) Combine $x^2$ terms: $3x^2 + 2x^2 = 5x^2$ Combine $x$ terms: $3x - 6x = -3x$ Combine constants: $-4 - 3 = -7$ So, (i) simplifies to: $$ 5x^2 - 3x - 7 $$ (ii) Combine $x^2$ terms: $x^2 + 4x^2 = 5x^2$ Combine $x$ terms: $2x - 5x = -3x$ Combine constants: $12 - 5 = 7$ So, (ii) simplifies to: $$ 5x^2 - 3x + 7 $$ (iii) Combine $x^2$ terms: $3x^2 + 2x^2 = 5x^2$ Combine $x$ terms: $-6x + 3x = -3x$ Combine constants: $3 - 10 = -7$ So, (iii) simplifies to: $$ 5x^2 - 3x - 7 $$ 8. **Step 2: Compare simplified forms**: (i) and (iii) are both $5x^2 - 3x - 7$, so they are equivalent. (ii) is $5x^2 - 3x + 7$, which differs in the constant term, so it is not equivalent to (i) or (iii). --- **Final answers:** - Sum of polynomials: $-4x^2 + 3y^2 - 3xy$ - Equivalent polynomials: (i) and (iii) are equivalent because they simplify to the same polynomial $5x^2 - 3x - 7$.