Which among the following is the correct value of $(x + a)(x + b)$?

### Concept & Formula This question requires basic polynomial expansion, commonly known as the FOIL method (First, Outer, Inner, Last). It is a fundamental algebraic identity used to multiply two binomials. $$ (x+a)(x+b) = x^2 + (a+b)x + ab $$ ### Step-by-Step Solution * **Given:** * Expression to expand: $(x + a)(x + b)$ * **Calculation:** 1. Multiply the First terms of each binomial: $x \cdot x = x^2$. 2. Multiply the Outer terms: $x \cdot b = bx$. 3. Multiply the Inner terms: $a \cdot x = ax$. 4. Multiply the Last terms of each binomial: $a \cdot b = ab$. 5. Combine all the results: $x^2 + bx + ax + ab$. 6. Group the like terms in the middle by factoring out the common variable $x$. 7. $bx + ax = (b + a)x = (a + b)x$. 8. Rewrite the final expression: $x^2 + (a + b)x + ab$. ### Exam Strategy & Shortcut This is a standard algebraic identity that should be memorized for competitive exams. However, if you forget it, use the substitution method. Let $x=1, a=2, b=3$. The expression $(1+2)(1+3) = (3)(4) = 12$. Now test the options with these values: (a) $1^2 + (2)(3)(1) + (2)(3) = 1 + 6 + 6 = 13$ (Incorrect) (c) $1^2 + (2+3)1 + (2)(3) = 1 + 5 + 6 = 12$ (Correct) This confirms the identity instantly. ### Common Pitfall A common error is rushing and confusing the sum and product of the constants $a$ and $b$, leading students to select an option like $x^2 + abx + ab$. Remember that the middle term is always the sum of the constants, and the final term is their product. ### Final Answer Therefore, the correct answer is $x^2 + (a + b)x + ab$.