To determine which expressions represent a quadratic expression in factored form, let's analyze each expression given:
1. [tex]\(x^2 - x - 72\)[/tex]: This is already in the expanded form of a quadratic expression and not in a factored form. So it is not a factored form.
2. [tex]\((x + 3)(x - 7)\)[/tex]: To check if this is a quadratic expression in factored form, let's expand it:
[tex]\[
(x + 3)(x - 7) = x(x - 7) + 3(x - 7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21
\][/tex]
This expression expands to a quadratic form, so [tex]\((x + 3)(x - 7)\)[/tex] is indeed a quadratic expression in factored form.
3. [tex]\(-8(x + 56)\)[/tex]: This expression can be expanded as follows:
[tex]\[
-8(x + 56) = -8x - 448
\][/tex]
This is a linear expression (not a quadratic expression), thus it is not in the factored form of a quadratic expression.
4. [tex]\((x + 1)(x - 2)\)[/tex]: To check if this is a quadratic expression in factored form, let's expand it:
[tex]\[
(x + 1)(x - 2) = x(x - 2) + 1(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2
\][/tex]
This expression expands to a quadratic form, so [tex]\((x + 1)(x - 2)\)[/tex] is indeed a quadratic expression in factored form.
5. [tex]\((x - 2) + (x + 3)\)[/tex]: This is simply the sum of two linear expressions:
[tex]\[
(x - 2) + (x + 3) = x - 2 + x + 3 = 2x + 1
\][/tex]
This is a linear expression (not a quadratic expression), hence it is not in the factored form of a quadratic expression.
Thus, the expressions that represent a quadratic expression in factored form are:
[tex]\[
\boxed{(x+3)(x-7), (x+1)(x-2)}
\][/tex]
