Mustafa was asked whether the following equation is an identity: $\newline$ $x(9 x+3)+3(x+3)=9(x+1)^{2}$ $\newline$ He performed the following steps: $\newline$ $x(9 x+3)+3(x+3)$ $\newline$ $\stackrel{\text { Step } 1}{\hookrightarrow}=9 x^{2}+6 x+9$ $\newline$ $\stackrel{\text { Step } 2}{\hookrightarrow}=(3 x+3)^{2}$ $\newline$ $\stackrel{\text { Step } 3}{\hookrightarrow}=(3(x+1))^{2}$ $\newline$ $\stackrel{\text { Step } 4}{\hookrightarrow}=9(x+1)^{2}$ $\newline$ For this reason, Mustafa stated that the equation is a true identity. $\newline$ Is Mustafa correct? If not, in which step did he make a mistake? $\newline$ Choose $1$ answer: $\newline$ (A) Mustafa is correct. $\newline$ (B) Mustafa is incorrect. He made a mistake in step $1$ . $\newline$ (C) Mustafa is incorrect. He made a mistake in step $2$ . $\newline$ (D) Mustafa is incorrect. He made a mistake in step $4$ .

Full solution

Q. Mustafa was asked whether the following equation is an identity:

\newline

x(9 x+3)+3(x+3)=9(x+1)^{2}

\newline

He performed the following steps:

\newline

x(9 x+3)+3(x+3)

\newline

\stackrel{\text { Step } 1}{\hookrightarrow}=9 x^{2}+6 x+9

\newline

\stackrel{\text { Step } 2}{\hookrightarrow}=(3 x+3)^{2}

\newline

\stackrel{\text { Step } 3}{\hookrightarrow}=(3(x+1))^{2}

\newline

\stackrel{\text { Step } 4}{\hookrightarrow}=9(x+1)^{2}

\newline

For this reason, Mustafa stated that the equation is a true identity.

\newline

Is Mustafa correct? If not, in which step did he make a mistake?

\newline

Choose

1

answer:

\newline

(A) Mustafa is correct.

\newline

(B) Mustafa is incorrect. He made a mistake in step

1

\newline

2

\newline

(D) Mustafa is incorrect. He made a mistake in step

4

Expand and Simplify: Check if the right side of the equation simplifies to the same expression. $\newline$ $9(x+1)^{2} = 9(x^2 + 2x + 1)$ $\newline$ Expand the squared term. $\newline$ $= 9x^2 + 18x + 9$
Compare Left and Right: Compare the left side from Step $1$ with the right side from Step $2$ . $\newline$ Left side: $9x^2 + 6x + 9$ $\newline$ Right side: $9x^2 + 18x + 9$ $\newline$ The two expressions are not equal because the coefficients of $x$ are different.