The result of the division of the polynomials is: [tex]x^2 - 2x - 3[/tex]
We have that:
[tex]\frac{x^3 - 6}{x + 2} = ax^2 + bx + c[/tex]
Applying cross multiplication:
[tex](ax^2 + bx + c)(x + 2) = x^3 - 6[/tex]
[tex]ax^3 + (2a + b)x^2 + (2b + c)x + 2c = x^3 - 6[/tex]
Equaling both sides, we have:
[tex]ax^3 = x^3 \rightarrow a = 1[/tex]
[tex]2c = -6 \rightarrow c = -3[/tex]
[tex]2a + b = 0 \rightarrow b = -2a = -2[/tex]
Thus, the result is:
[tex]ax^2 + bx + c = x^2 - 2x - 3[/tex]
A similar problem is given at https://brainly.com/question/24662212
