To factorize the polynomial [tex]\( 4x^2 + 23x - 72 \)[/tex], we need to identify the correct factored form among the given options.
Let's examine each option to determine which one properly reconstitutes the given polynomial when expanded:
1. [tex]\((4x + 8)(x - 9)\)[/tex]:
[tex]\[
(4x + 8)(x - 9) = 4x \cdot x + 4x \cdot (-9) + 8 \cdot x + 8 \cdot (-9) \\
= 4x^2 - 36x + 8x - 72 \\
= 4x^2 - 28x - 72
\][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
2. [tex]\((4x - 8)(x + 9)\)[/tex]:
[tex]\[
(4x - 8)(x + 9) = 4x \cdot x + 4x \cdot 9 + (-8) \cdot x + (-8) \cdot 9 \\
= 4x^2 + 36x - 8x - 72 \\
= 4x^2 + 28x - 72
\][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
3. [tex]\((4x + 9)(x - 8)\)[/tex]:
[tex]\[
(4x + 9)(x - 8) = 4x \cdot x + 4x \cdot (-8) + 9 \cdot x + 9 \cdot (-8) \\
= 4x^2 - 32x + 9x - 72 \\
= 4x^2 - 23x - 72
\][/tex]
This does not match [tex]\(4x^2 + 23x - 72\)[/tex].
4. [tex]\((4x - 9)(x + 8)\)[/tex]:
[tex]\[
(4x - 9)(x + 8) = 4x \cdot x + 4x \cdot 8 + (-9) \cdot x + (-9) \cdot 8 \\
= 4x^2 + 32x - 9x - 72 \\
= 4x^2 + 23x - 72
\][/tex]
This matches [tex]\(4x^2 + 23x - 72\)[/tex].
Therefore, the factored form of the polynomial [tex]\( 4x^2 + 23x - 72 \)[/tex] is [tex]\((4x - 9)(x + 8)\)[/tex]. Thus, it matches the fourth option provided in the question.
