To find which expressions correspond to the given options A, B, C, and D, we need to expand and simplify each expression step by step.
1. Expression: [tex]\((x^2 + 15x + 65) + (2x - 5)(3x + 8)\)[/tex]
First, expand [tex]\((2x - 5)(3x + 8)\)[/tex]:
[tex]\[
(2x - 5)(3x + 8) = 2x \cdot 3x + 2x \cdot 8 - 5 \cdot 3x - 5 \cdot 8 = 6x^2 + 16x - 15x - 40 = 6x^2 + x - 40
\][/tex]
Now, add this result to [tex]\(x^2 + 15x + 65\)[/tex]:
[tex]\[
x^2 + 15x + 65 + 6x^2 + x - 40 = 7x^2 + 16x + 25
\][/tex]
The equivalent expression is B.
2. Expression: [tex]\((4x + 1)(3x - 4) - (5x^2 - 10x - 12)\)[/tex]
First, expand [tex]\((4x + 1)(3x - 4)\)[/tex]:
[tex]\[
(4x + 1)(3x - 4) = 4x \cdot 3x + 4x \cdot (-4) + 1 \cdot 3x + 1 \cdot (-4) = 12x^2 - 16x + 3x - 4 = 12x^2 - 13x - 4
\][/tex]
Now, subtract [tex]\(5x^2 - 10x - 12\)[/tex] from this result:
[tex]\[
12x^2 - 13x - 4 - 5x^2 + 10x + 12 = 7x^2 - 3x + 8
\][/tex]
The equivalent expression is D.
3. Expression: [tex]\((8x^2 + 19x + 4) + (3x + 2)(x - 5)\)[/tex]
First, expand [tex]\((3x + 2)(x - 5)\)[/tex]:
[tex]\[
(3x + 2)(x - 5) = 3x \cdot x + 3x \cdot (-5) + 2 \cdot x + 2 \cdot (-5) = 3x^2 - 15x + 2x - 10 = 3x^2 - 13x - 10
\][/tex]
Now, add this result to [tex]\(8x^2 + 19x + 4\)[/tex]:
[tex]\[
8x^2 + 19x + 4 + 3x^2 - 13x - 10 = 11x^2 + 6x - 6
\][/tex]
The equivalent expression is A.
4. Expression: [tex]\((6x + 1)(3x - 7) - (7x^2 - 34x - 20)\)[/tex]
First, expand [tex]\((6x + 1)(3x - 7)\)[/tex]:
[tex]\[
(6x + 1)(3x - 7) = 6x \cdot 3x + 6x \cdot (-7) + 1 \cdot 3x + 1 \cdot (-7) = 18x^2 - 42x + 3x - 7 = 18x^2 - 39x - 7
\][/tex]
Now, subtract [tex]\(7x^2 - 34x - 20\)[/tex] from this result:
[tex]\[
18x^2 - 39x - 7 - 7x^2 + 34x + 20 = 11x^2 - 5x + 13
\][/tex]
The equivalent expression is C.
Final Answer:
[tex]\[
\begin{align*}
(x^2 + 15x + 65) + (2x - 5)(3x + 8) & \text{ is equivalent to expression } \boxed{B} \\
(4x + 1)(3x - 4) - (5x^2 - 10x - 12) & \text{ is equivalent to expression } \boxed{D} \\
(8x^2 + 19x + 4) + (3x + 2)(x - 5) & \text{ is equivalent to expression } \boxed{A} \\
(6x + 1)(3x - 7) - (7x^2 - 34x - 20) & \text{ is equivalent to expression } \boxed{C} \\
\end{align*}
\][/tex]
