Answer :
To factor the trinomial [tex]\( 24x^2 - 49x - 40 \)[/tex], we start by examining the possible pairs of factors.
Consider the general form of the factored trinomial:
[tex]\[ (ax + b)(cx + d) \][/tex]
where [tex]\( a \cdot c = 24 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]) and [tex]\( b \cdot d = -40 \)[/tex] (the constant term). Also, the cross-terms need to add up to give the middle term, [tex]\(-49x\)[/tex].
Let's evaluate each choice to see if it correctly factors the trinomial.
### Choice A: [tex]\((4x - 8)(6x + 5)\)[/tex]
Let's expand [tex]\((4x - 8)(6x + 5)\)[/tex]:
[tex]\[ (4x - 8)(6x + 5) = 4x \cdot 6x + 4x \cdot 5 - 8 \cdot 6x - 8 \cdot 5 \\ = 24x^2 + 20x - 48x - 40 \\ = 24x^2 - 28x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice B: [tex]\((4x + 8)(6x - 5)\)[/tex]
Let's expand [tex]\((4x + 8)(6x - 5)\)[/tex]:
[tex]\[ (4x + 8)(6x - 5) = 4x \cdot 6x + 4x \cdot (-5) + 8 \cdot 6x + 8 \cdot (-5) \\ = 24x^2 - 20x + 48x - 40 \\ = 24x^2 + 28x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice C: [tex]\((3x + 8)(8x - 5)\)[/tex]
Let's expand [tex]\((3x + 8)(8x - 5)\)[/tex]:
[tex]\[ (3x + 8)(8x - 5) = 3x \cdot 8x + 3x \cdot (-5) + 8 \cdot 8x + 8 \cdot (-5) \\ = 24x^2 - 15x + 64x - 40 \\ = 24x^2 + 49x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice D: [tex]\((3x - 8)(8x + 5)\)[/tex]
Let's expand [tex]\((3x - 8)(8x + 5)\)[/tex]:
[tex]\[ (3x - 8)(8x + 5) = 3x \cdot 8x + 3x \cdot 5 - 8 \cdot 8x - 8 \cdot 5 \\ = 24x^2 + 15x - 64x - 40 \\ = 24x^2 - 49x - 40 \][/tex]
This matches [tex]\( 24x^2 - 49x - 40 \)[/tex].
Therefore, the correct factorization of the trinomial [tex]\( 24x^2 - 49x - 40 \)[/tex] is:
[tex]\[ (3x - 8)(8x + 5) \][/tex]
Thus, the answer is:
[tex]\[ \boxed{D} \][/tex]
Consider the general form of the factored trinomial:
[tex]\[ (ax + b)(cx + d) \][/tex]
where [tex]\( a \cdot c = 24 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]) and [tex]\( b \cdot d = -40 \)[/tex] (the constant term). Also, the cross-terms need to add up to give the middle term, [tex]\(-49x\)[/tex].
Let's evaluate each choice to see if it correctly factors the trinomial.
### Choice A: [tex]\((4x - 8)(6x + 5)\)[/tex]
Let's expand [tex]\((4x - 8)(6x + 5)\)[/tex]:
[tex]\[ (4x - 8)(6x + 5) = 4x \cdot 6x + 4x \cdot 5 - 8 \cdot 6x - 8 \cdot 5 \\ = 24x^2 + 20x - 48x - 40 \\ = 24x^2 - 28x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice B: [tex]\((4x + 8)(6x - 5)\)[/tex]
Let's expand [tex]\((4x + 8)(6x - 5)\)[/tex]:
[tex]\[ (4x + 8)(6x - 5) = 4x \cdot 6x + 4x \cdot (-5) + 8 \cdot 6x + 8 \cdot (-5) \\ = 24x^2 - 20x + 48x - 40 \\ = 24x^2 + 28x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice C: [tex]\((3x + 8)(8x - 5)\)[/tex]
Let's expand [tex]\((3x + 8)(8x - 5)\)[/tex]:
[tex]\[ (3x + 8)(8x - 5) = 3x \cdot 8x + 3x \cdot (-5) + 8 \cdot 8x + 8 \cdot (-5) \\ = 24x^2 - 15x + 64x - 40 \\ = 24x^2 + 49x - 40 \][/tex]
This does not match [tex]\( 24x^2 - 49x - 40 \)[/tex].
### Choice D: [tex]\((3x - 8)(8x + 5)\)[/tex]
Let's expand [tex]\((3x - 8)(8x + 5)\)[/tex]:
[tex]\[ (3x - 8)(8x + 5) = 3x \cdot 8x + 3x \cdot 5 - 8 \cdot 8x - 8 \cdot 5 \\ = 24x^2 + 15x - 64x - 40 \\ = 24x^2 - 49x - 40 \][/tex]
This matches [tex]\( 24x^2 - 49x - 40 \)[/tex].
Therefore, the correct factorization of the trinomial [tex]\( 24x^2 - 49x - 40 \)[/tex] is:
[tex]\[ (3x - 8)(8x + 5) \][/tex]
Thus, the answer is:
[tex]\[ \boxed{D} \][/tex]
