Certainly! Let's simplify the given expression step-by-step:
We start with:
[tex]\[ 2(x + 2)^2 + 2(x + 2) - 1 \][/tex]
Step 1: Expand the squared term:
First, expand [tex]\((x + 2)^2\)[/tex]:
[tex]\[
(x + 2)^2 = (x + 2)(x + 2) = x^2 + 4x + 4
\][/tex]
Step 2: Multiply the expanded term by 2:
Now multiply the result by 2:
[tex]\[
2(x^2 + 4x + 4) = 2x^2 + 8x + 8
\][/tex]
Step 3: Expand the second term:
Next, expand [tex]\(2(x + 2)\)[/tex]:
[tex]\[
2(x + 2) = 2x + 4
\][/tex]
Step 4: Combine all expanded terms:
Combine all the terms from the previous expansions and include the constant [tex]\(-1\)[/tex]:
[tex]\[
2x^2 + 8x + 8 + 2x + 4 - 1
\][/tex]
Step 5: Simplify the combined expression:
Combine like terms:
[tex]\[
2x^2 + (8x + 2x) + (8 + 4 - 1)
\][/tex]
[tex]\[
2x^2 + 10x + 11
\][/tex]
So, the simplified polynomial form of the given expression [tex]\(2(x + 2)^2 + 2(x + 2) - 1\)[/tex] is:
[tex]\[
\boxed{2x^2 + 10x + 11}
\][/tex]
