To simplify the polynomial expression [tex]\((2x - 1)(2x^2 + 5x + 3) + (3x + 6)\)[/tex], we can break it down step by step.
### Step 1: Expand the polynomial
First, we expand [tex]\((2x - 1)(2x^2 + 5x + 3)\)[/tex].
[tex]\[
\begin{align*}
(2x - 1)(2x^2 + 5x + 3) &= 2x \cdot (2x^2 + 5x + 3) - 1 \cdot (2x^2 + 5x + 3) \\
&= (2x \cdot 2x^2) + (2x \cdot 5x) + (2x \cdot 3) - (1 \cdot 2x^2) - (1 \cdot 5x) - (1 \cdot 3) \\
&= 4x^3 + 10x^2 + 6x - 2x^2 - 5x - 3
\end{align*}
\][/tex]
Next, we combine the like terms in the resulting polynomial:
[tex]\[
4x^3 + (10x^2 - 2x^2) + (6x - 5x) - 3 \\
= 4x^3 + 8x^2 + x - 3
\][/tex]
### Step 2: Add the remaining polynomial term
Now, add [tex]\((3x + 6)\)[/tex] to the expanded result:
[tex]\[
(4x^3 + 8x^2 + x - 3) + (3x + 6) \\
= 4x^3 + 8x^2 + x + 3x - 3 + 6
\][/tex]
Again, combine like terms:
[tex]\[
= 4x^3 + 8x^2 + (x + 3x) + (-3 + 6) \\
= 4x^3 + 8x^2 + 4x + 3
\][/tex]
### Simplified Polynomial Expression
After simplifying the given polynomial expression step by step, we arrive at the result:
[tex]\[
4x^3 + 8x^2 + 4x + 3
\][/tex]
