[tex]a^3+b^3=(a+b)(a^2-ab+b^2)\\\\a^3-b^3=(a-b)(a^2+ab+b^2)\\\\(x-4)(x^2+4x\underline{-16})\qquad \mathbb{NOT}\\\\(x-1)(x^2\underline{-x}+1)\qquad\mathbb{NOT}\\\\(x-1)(x^2+x+1)=(x-1)(x^2+(x)(1)+1^2)\qquad\mathbb{YES}\\\\(x+1)(x^2+x\underline{-1})\qquad\mathbb{NOT}\\\\(x+4)(x^2-4x+16)=(x+4)(x^2+(x)(4)+4^2)\qquad\mathbb{YES}\\\\(x+4)(x^2\underline{+4x}+4)\qquad\mathbb{NOT}[/tex]
The expressions that result in a sum or difference of cubes are:
[tex]\mathbf{(x - 1)(x^2 + x + 1)}[/tex] and [tex]\mathbf{(x + 4)(x^2 - 4x + 16) }[/tex]
The sum of cubes is represented as:
[tex]\mathbf{x^3 + y^3 = (x + y)(x^2 -xy + y^2)}[/tex]
The difference of cubes is represented as:
[tex]\mathbf{x^3 - y^3 = (x - y)(x^2 +xy + y^2)}[/tex]
Next, we test the options
[tex]\mathbf{(x - 4)(x^2 + 4x - 16)}[/tex]
Compare to [tex]\mathbf{x^3 - y^3 = (x - y)(x^2 +xy + y^2)}[/tex].
[tex]\mathbf{(x - 4)(x^2 + 4x - 16)}[/tex] does not represent the sum or difference of cubes
[tex]\mathbf{(x - 1)(x^2 - x + 1)}[/tex]
Compare to [tex]\mathbf{x^3 - y^3 = (x - y)(x^2 +xy + y^2)}[/tex].
[tex]\mathbf{(x - 1)(x^2 - x + 1)}[/tex] does not represent the sum or difference of cubes
[tex]\mathbf{(x - 1)(x^2 + x + 1)}[/tex]
Compare to [tex]\mathbf{x^3 - y^3 = (x - y)(x^2 +xy + y^2)}[/tex].
[tex]\mathbf{(x - 1)(x^2 + x + 1)}[/tex] represents the difference of cubes
[tex]\mathbf{(x + 4)(x^2 - 4x + 16) }[/tex]
Compare to [tex]\mathbf{x^3 + y^3 = (x + y)(x^2 -xy + y^2)}[/tex]
[tex]\mathbf{(x + 4)(x^2 - 4x + 16) }[/tex] represents the sum of cubes
[tex]\mathbf{(x + 4)(x^2 + 4x + 16)}[/tex]
Compare to [tex]\mathbf{x^3 + y^3 = (x + y)(x^2 -xy + y^2)}[/tex]
[tex]\mathbf{(x + 4)(x^2 + 4x + 16)}[/tex] does not represent the sum of cubes
Hence, the expressions that result in a sum or difference of cubes are:
[tex]\mathbf{(x - 1)(x^2 + x + 1)}[/tex] and [tex]\mathbf{(x + 4)(x^2 - 4x + 16) }[/tex]
Read more about sums and difference of cubes at:
https://brainly.com/question/17077929
