If the HCF of the expressions \((x+3)(2x^2 - 3x + a) \text{ and } (x-2)(3x^2 + 10x - b) \text{ is } x^2 + x - 6\) then what is the value of (2a - 3b) ?

Given:

Expression 1: (x+3)(2x² - 3x + a)

Expression 2: (x-2)(3x² + 10x - b)

HCF of Expression 1 and Expression 2 is x² + x - 6

Formula used:

If an expression is the HCF of two polynomials, then it must be a factor of both polynomials.

This means that the roots of the HCF polynomial must also be roots of the two given expressions.

Calculation:

First, factorize the HCF: x² + x - 6

⇒ x² + 3x - 2x - 6

⇒ x(x + 3) - 2(x + 3)

⇒ (x + 3)(x - 2)

Since (x + 3)(x - 2) is the HCF, it must be a factor of both given expressions.

Consider Expression 1: (x + 3)(2x² - 3x + a)

We already have the factor (x + 3). For the HCF to be (x + 3)(x - 2), it means that (x - 2) must be a factor of (2x² - 3x + a).

If (x - 2) is a factor of (2x² - 3x + a), then substituting x = 2 into (2x² - 3x + a) should result in 0.

⇒ 2(2)² - 3(2) + a = 0

⇒ 2(4) - 6 + a = 0

⇒ 8 - 6 + a = 0

⇒ 2 + a = 0

⇒ a = -2

Consider Expression 2: (x - 2)(3x² + 10x - b)

We already have the factor (x - 2). For the HCF to be (x + 3)(x - 2), it means that (x + 3) must be a factor of (3x² + 10x - b).

If (x + 3) is a factor of (3x² + 10x - b), then substituting x = -3 into (3x² + 10x - b) should result in 0.

⇒ 3(-3)² + 10(-3) - b = 0

⇒ 3(9) - 30 - b = 0

⇒ 27 - 30 - b = 0

⇒ -3 - b = 0

⇒ b = -3

Now, we need to find the value of (2a - 3b):

⇒ 2a - 3b = 2(-2) - 3(-3)

⇒ 2a - 3b = -4 - (-9)

⇒ 2a - 3b = -4 + 9

⇒ 2a - 3b = 5

∴ The correct answer is option 3.