Expand and simplify (3x + 2)(2x - 5) into the form ax^2 + kx + n. What is the value of the expression a - n + k?

Introduction / Context:
This is an algebra question involving expansion of a product of two binomials and identification of coefficients in the resulting quadratic expression. It checks familiarity with the distributive property and standard algebraic multiplication. Once the expression is written in the form ax^2 + kx + n, we are asked to compute a derived value, namely a - n + k. This type of question is common in algebra sections of aptitude exams.

Given Data / Assumptions:

• The expression to expand is (3x + 2)(2x - 5).
• After expansion, it can be written in the form ax^2 + kx + n.
• a is the coefficient of x^2, k is the coefficient of x, and n is the constant term.
• We must find the value of a - n + k.

Concept / Approach:
We use the distributive property or the FOIL method to expand the product of two binomials. After expansion, we collect like terms in x^2, x and the constant term to match the standard quadratic form. Once the coefficients a, k and n are identified, we substitute them into the expression a - n + k and simplify to get the numeric answer.

Step-by-Step Solution:
Step 1: Start with the given expression (3x + 2)(2x - 5). Step 2: Multiply each term in the first bracket by each term in the second bracket. Step 3: First term: 3x * 2x = 6x^2. Step 4: Outer term: 3x * (−5) = −15x. Step 5: Inner term: 2 * 2x = 4x. Step 6: Last term: 2 * (−5) = −10. Step 7: Combine the like terms for x: −15x + 4x = −11x. Step 8: So the expanded expression is 6x^2 − 11x − 10. Step 9: Comparing with ax^2 + kx + n, we identify a = 6, k = −11 and n = −10. Step 10: Compute a − n + k = 6 − (−10) + (−11). Step 11: Simplify: 6 − (−10) = 6 + 10 = 16, and 16 + (−11) = 5. Step 12: Therefore, a − n + k = 5.

Verification / Alternative check:
Recheck the expansion: (3x + 2)(2x − 5) = 3x * 2x + 3x * (−5) + 2 * 2x + 2 * (−5) = 6x^2 − 15x + 4x − 10 = 6x^2 − 11x − 10. The coefficients are therefore confirmed as a = 6, k = −11 and n = −10. Substituting into a − n + k gives 6 − (−10) + (−11) = 6 + 10 − 11 = 16 − 11 = 5, confirming the answer.

Why Other Options Are Wrong:
Options (b) 8, (c) 9, and (d) 10 would correspond to miscalculations, either from incorrect expansion or incorrect handling of negative signs when computing a − n + k. Since the verified result is 5, these options do not match the correct computation.

Common Pitfalls:
Common errors include forgetting one of the terms in the FOIL expansion, adding −15x and 4x incorrectly, or misinterpreting the constant term when substituting into a − n + k. The negative signs in k and n can easily be mishandled. Carefully writing each intermediate step and paying attention to signs helps avoid these mistakes.

Final Answer:
The value of a − n + k is 5.