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Alaina was asked whether the following equation is an identity:

(2x+1)^(2)-(x+1)^(2)=3(x+1)^(2)-1
She performed the following steps:

(2x+1)^(2)-(x+1)^(2)

rarr"" Step "1"=4x^(2)+4x+1-x^(2)+2x+1

longrightarrow^(" Step "2)=3x^(2)+6x+2

rarr"" Step "3"=3x^(2)+6x+3-1

longrightarrow^(" Step "4)=3(x^(2)+2x+1)-1

^(" Step "5)=3(x+1)^(2)-1
For this reason, Alaina stated that the equation is a true identity.
Is Alaina correct? If not, in which step did she make a mistake?
Choose 1 answer:
(A) Alaina is correct.
(B) Alaina is incorrect. She made a mistake in step 1.
(C) Alaina is incorrect. She made a mistake in step 3.
(D) Alaina is incorrect. She made a mistake in step 5.

Alaina was asked whether the following equation is an identity:\newline(2x+1)2(x+1)2=3(x+1)21 (2 x+1)^{2}-(x+1)^{2}=3(x+1)^{2}-1 \newlineShe performed the following steps:\newline(2x+1)2(x+1)2 (2 x+1)^{2}-(x+1)^{2} \newline Step 1=4x2+4x+1x2+2x+1 \stackrel{\text { Step } 1}{\hookrightarrow}=4 x^{2}+4 x+1-x^{2}+2 x+1 \newline Step 2=3x2+6x+2 \stackrel{\text { Step } 2}{\hookrightarrow}=3 x^{2}+6 x+2 \newline Step 3=3x2+6x+31 \stackrel{\text { Step } 3}{\hookrightarrow}=3 x^{2}+6 x+3-1 \newline Step 4=3(x2+2x+1)1 \stackrel{\text { Step } 4}{\hookrightarrow}=3\left(x^{2}+2 x+1\right)-1 \newline Step 5=3(x+1)21 \stackrel{\text { Step } 5}{\hookrightarrow}=3(x+1)^{2}-1 \newlineFor this reason, Alaina stated that the equation is a true identity.\newlineIs Alaina correct? If not, in which step did she make a mistake?\newlineChoose 11 answer:\newline(A) Alaina is correct.\newline(B) Alaina is incorrect. She made a mistake in step 11.\newline(C) Alaina is incorrect. She made a mistake in step 33.\newline(D) Alaina is incorrect. She made a mistake in step 55.

Full solution

Q. Alaina was asked whether the following equation is an identity:\newline(2x+1)2(x+1)2=3(x+1)21 (2 x+1)^{2}-(x+1)^{2}=3(x+1)^{2}-1 \newlineShe performed the following steps:\newline(2x+1)2(x+1)2 (2 x+1)^{2}-(x+1)^{2} \newline Step 1=4x2+4x+1x2+2x+1 \stackrel{\text { Step } 1}{\hookrightarrow}=4 x^{2}+4 x+1-x^{2}+2 x+1 \newline Step 2=3x2+6x+2 \stackrel{\text { Step } 2}{\hookrightarrow}=3 x^{2}+6 x+2 \newline Step 3=3x2+6x+31 \stackrel{\text { Step } 3}{\hookrightarrow}=3 x^{2}+6 x+3-1 \newline Step 4=3(x2+2x+1)1 \stackrel{\text { Step } 4}{\hookrightarrow}=3\left(x^{2}+2 x+1\right)-1 \newline Step 5=3(x+1)21 \stackrel{\text { Step } 5}{\hookrightarrow}=3(x+1)^{2}-1 \newlineFor this reason, Alaina stated that the equation is a true identity.\newlineIs Alaina correct? If not, in which step did she make a mistake?\newlineChoose 11 answer:\newline(A) Alaina is correct.\newline(B) Alaina is incorrect. She made a mistake in step 11.\newline(C) Alaina is incorrect. She made a mistake in step 33.\newline(D) Alaina is incorrect. She made a mistake in step 55.
  1. Expand Expression: Expand (2x+1)2(2x+1)^{2} and (x+1)2(x+1)^{2}.\newline (2x+1)2=4x2+4x+1(2x+1)^{2} = 4x^{2} + 4x + 1\newline (x+1)2=x2+2x+1(x+1)^{2} = x^{2} + 2x + 1
  2. Subtract Squares: Subtract (x+1)2(x+1)^{2} from (2x+1)2(2x+1)^{2}.\newline4x2+4x+1(x2+2x+1)=3x2+2x4x^{2} + 4x + 1 - (x^{2} + 2x + 1) = 3x^{2} + 2x

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