If ( x + 2) and (x - 3) are the factors of $x^2 +k_1x + k_2$, then :

$k_1 = 1$ and $k_2 = -6$

If ( x - 2) and (x + 3) are the factors of $x^2 +k_1x + k_2$ then,

Putting x = 2 in the given equation, we get

= 2² + 2k₁ + k₂ = 0

⇒ 2k₁ + k₂ = - 4      …..(A)

Similarly, Putting x = -3 in the given equation,

= (-3) ² - 3k₁ + k₂ = 0

= - 3k₁ + k₂ = -9      ……(B)

Subtracting (A) and (B)

= 5k₁ = 5

= k₁ = 1

Putting value of k₁ in (A)

= k₂ = -6

k₁ = 1 and k₂ = -6