If (x + 2) and (x + 3) are factors of x³ + ax + b, find the values of a and b.

Let f(x) = x³ + ax + b

Since (x + 2) and (x + 3) are factors, by the factor theorem, f(−2) = 0 and f(−3) = 0.

⇒ f(-2) = 0

⇒ (-2)³ + a(-2) + b = 0

⇒ -8 - 2a + b = 0

⇒ -2a + b = 8 ….(1)

⇒ f(-3) = 0

⇒ (-3)³ + a(-3) + b = 0

⇒ -27 - 3a + b = 0

⇒ -3a + b = 27 ….(2)

Subtract equation (2) from equation (1), we get:

⇒ -2a + b - (-3a + b) = 8 - 27

⇒ -2a + 3a = -19

⇒ a = -19

Substituting value of a in equation (1), we get :

⇒ -2(-19) + b = 8

⇒ 38 + b = 8

⇒ b = 8 - 38

⇒ b = -30

Hence, the value of a = -19 and b = -30.