Consider the cubic polynomial,
[tex](x+a)(x+b)(x+c)[/tex]
Expanding this gives
[tex]x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=x^3+x^2-4x-4[/tex]
We can factor this by grouping,
[tex]x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x^2-4)(x+1)=(x-2)(x+2)(x+1)[/tex]
Then letting [tex]a=-2[/tex], [tex]b=2[/tex], and [tex]c=1[/tex] gives [tex]a^3+b^3+c^3=-8+8+1=\boxed1[/tex]
