[tex]\text{Let } f(x) = x^3 + 2x^2 + kx - 6[/tex]
As (x + 1) is a factor, it follows that
[tex]f(-1) = 0[/tex]
[tex]\implies (-1)^3 + 2(-1)^2 + k(-1) - 6 = 0[/tex]
[tex]-1 + 2 -k - 6 = 0[/tex]
[tex]-k - 5 = 0[/tex]
[tex]k = -5[/tex]
[tex]\implies f(x) = x^3 + 2x^2 - 5x - 6[/tex]
Now we can use long division to find what is left of f(x) after it is divided by (x + 1). (Apologies, this is the best way I can represent long division on Brainly at this current time - I hope it's clear)
x^2 + x - 6
x + 1 ( x^3 + 2x^2 - 5x - 6
x^3 + x^2
x^2 - 5x
x^2 + x
-6x - 6
-6x - 6
0
So the remainder when f(x) is divided by (x + 1) is
[tex]x^2 + x - 6[/tex]
Factorising this we get
[tex](x + 3)(x - 2)[/tex]
So the three factors of f(x) are (x + 1), (x + 3) and (x - 2).
