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Using arithmetic operations, rewrite this polynomial in standard, simplest form.
[tex]\[ \left(3x^2 + 9x + 6\right) - \left(8x^2 + 3x - 10\right) + (2x + 4)(3x - 7) \][/tex]

A. \( x^2 + 4x - 12 \)

B. \( x^2 + 8x - 42 \)

C. \( 17x^2 + 10x - 32 \)

D. [tex]\( 17x^2 + 14x + 24 \)[/tex]

Sagot :

GinnyAnsweridn
To rewrite the given polynomial in its simplest form, we'll proceed through the following steps:

1. Expand the terms inside the expression:

Given expression:
[tex]\[ \left(3x^2 + 9x + 6\right) - \left(8x^2 + 3x - 10\right) + \left(2x + 4\right)\left(3x - 7\right) \][/tex]

2. Distribute and combine like terms:

First, distribute the subtraction:
[tex]\[ \left(3x^2 + 9x + 6\right) - 8x^2 - 3x + 10 + \left(2x + 4\right)\left(3x - 7\right) \][/tex]

3. Expand the product:

Next, expand \(\left(2x + 4\right)\left(3x - 7\right):
[tex]\[ (2x \cdot 3x) + (2x \cdot -7) + (4 \cdot 3x) + (4 \cdot -7) \][/tex]
[tex]\[ = 6x^2 - 14x + 12x - 28 \][/tex]
[tex]\[ = 6x^2 - 2x - 28 \][/tex]

4. Substitute back into the expression:

Now place the expanded product back into the original expression:
[tex]\[ 3x^2 + 9x + 6 - 8x^2 - 3x + 10 + 6x^2 - 2x - 28 \][/tex]

5. Combine like terms:

Combine the \(x^2\) terms:
[tex]\[ 3x^2 - 8x^2 + 6x^2 = (3 - 8 + 6)x^2 = 1x^2 \][/tex]

Combine the \(x\) terms:
[tex]\[ 9x - 3x - 2x = (9 - 3 - 2)x = 4x \][/tex]

Combine the constant terms:
[tex]\[ 6 + 10 - 28 = 6 + 10 - 28 = -12 \][/tex]

6. Write the final expression:

Combine all like terms to get the final simplified polynomial:
[tex]\[ x^2 + 4x - 12 \][/tex]

Therefore, the correct answer is:
[tex]\( \boxed{x^2 + 4x - 12} \)[/tex]
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