1. Let's state the problem: Verify if the expression $ax^2 - bx^2 - bx + ax - a + b$ equals $(ax - bx)(x + 1) - a + b$. 2. Start by simplifying the left-hand side (LHS): $$ax^2 - bx^2 - bx + ax - a + b$$ Group like terms: $$ (ax^2 - bx^2) + (ax - bx) + (-a + b) $$ Factor where possible: $$ x^2(a - b) + x(a - b) + (-a + b) $$ 3. Now simplify the right-hand side (RHS): $$(ax - bx)(x + 1) - a + b$$ Factor inside the first parentheses: $$ (a - b)x (x + 1) - a + b $$ Expand: $$ (a - b)(x^2 + x) - a + b $$ Distribute: $$ (a - b)x^2 + (a - b)x - a + b $$ 4. Compare LHS and RHS: LHS: $$ x^2(a - b) + x(a - b) + (-a + b) $$ RHS: $$ (a - b)x^2 + (a - b)x - a + b $$ They are identical. 5. Conclusion: The given equality is correct.