(3x + 2) (2x - 5) = ax2 + kx + n. What is the value of a - n + k ?

Concept: To compare coefficients in an identity of the form (linear expression) × (linear expression) = ax2 + kx + n, fully expand the left side using the distributive (FOIL) method, then match the coefficients of x2, x, and the constant term on both sides — the x2 coefficient is a, the x coefficient is k, and the constant term is n.

Expand (3x + 2)(2x − 5):

1. 3x × 2x = 6x2

2. 3x × (−5) = −15x

3. 2 × 2x = 4x

4. 2 × (−5) = −10

Combining like terms: 6x2 + (−15x + 4x) − 10 = 6x2 − 11x − 10. Comparing with ax2 + kx + n gives a = 6, k = −11, and n = −10.

Substituting into a − n + k: 6 − (−10) + (−11) = 6 + 10 − 11 = 5.

Cross-check: Substitute x = 1 into both sides of the original identity: the left side gives (3 + 2)(2 − 5) = 5 × (−3) = −15, and the right side gives a + k + n = 6 − 11 − 10 = −15. The two sides match, confirming the expansion is correct, so a − n + k = 5.