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]
