To factor the polynomial [tex]\(4x^2 + 23x - 72\)[/tex], we'll start by identifying which of the given options correctly represent the polynomial in its factored form. Let's analyze the options step by step.
Given options are:
1. [tex]\((4x + 8)(x - 9)\)[/tex]
2. [tex]\((4x - 8)(x + 9)\)[/tex]
3. [tex]\((4x + 9)(x - 8)\)[/tex]
4. [tex]\((4x - 9)(x + 8)\)[/tex]
We are trying to determine which one of these is the correct factored form of the polynomial [tex]\(4x^2 + 23x - 72\)[/tex].
Now let's verify the correctness of each option by expanding them:
### Option 1: [tex]\((4x + 8)(x - 9)\)[/tex]
[tex]\[ (4x + 8)(x - 9) \][/tex]
[tex]\[ = 4x(x - 9) + 8(x - 9) \][/tex]
[tex]\[ = 4x^2 - 36x + 8x - 72 \][/tex]
[tex]\[ = 4x^2 - 28x - 72 \][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
### Option 2: [tex]\((4x - 8)(x + 9)\)[/tex]
[tex]\[ (4x - 8)(x + 9) \][/tex]
[tex]\[ = 4x(x + 9) - 8(x + 9) \][/tex]
[tex]\[ = 4x^2 + 36x - 8x - 72 \][/tex]
[tex]\[ = 4x^2 + 28x - 72 \][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
### Option 3: [tex]\((4x + 9)(x - 8)\)[/tex]
[tex]\[ (4x + 9)(x - 8) \][/tex]
[tex]\[ = 4x(x - 8) + 9(x - 8) \][/tex]
[tex]\[ = 4x^2 - 32x + 9x - 72 \][/tex]
[tex]\[ = 4x^2 - 23x - 72 \][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
### Option 4: [tex]\((4x - 9)(x + 8)\)[/tex]
[tex]\[ (4x - 9)(x + 8) \][/tex]
[tex]\[ = 4x(x + 8) - 9(x + 8) \][/tex]
[tex]\[ = 4x^2 + 32x - 9x - 72 \][/tex]
[tex]\[ = 4x^2 + 23x - 72 \][/tex]
This matches the original polynomial [tex]\(4x^2 + 23x - 72\)[/tex].
Therefore, the correct factored form of the polynomial [tex]\(4x^2 + 23x - 72\)[/tex] is:
[tex]\[\boxed{(4x - 9)(x + 8)}\][/tex]
And this corresponds to option 4.
