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Mabel was asked whether the following equation is an identity:

(4x-3)(x-2)^(2)=4x^(3)-19x^(2)+4x-12
She performed the following steps:

(4x-3)(x-2)^(2)

↪^(" Step "1)=(4x-3)(x-2)(x-2)

↪^(" Step ")=(4x-3)(x^(2)-2x-2x+4)

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

rarr"" Step "4"=4x^(3)-16x^(2)+16 x-3x^(2)-12 x-12

rarr"" Step "5"=4x^(3)-19x^(2)+4x-12
For this reason, Mabel stated that the equation is a true identity.
Is Mabel correct? If not, in which step did she make a mistake?
Choose 1 answer:
(A) Mabel is correct.
(B) Mabel is incorrect. She made a mistake in step 2.
(C) Mabel is incorrect. She made a mistake in step 3.
(D) Mabel is incorrect. She made a mistake in step 4.

Mabel was asked whether the following equation is an identity:\newline(4x3)(x2)2=4x319x2+4x12 (4 x-3)(x-2)^{2}=4 x^{3}-19 x^{2}+4 x-12 \newlineShe performed the following steps:\newline(4x3)(x2)2 (4 x-3)(x-2)^{2} \newline Step 1=(4x3)(x2)(x2) \stackrel{\text { Step } 1}{\hookrightarrow}=(4 x-3)(x-2)(x-2) \newline Step 2=(4x3)(x22x2x+4) \stackrel{\text { Step } 2}{\hookrightarrow}=(4 x-3)\left(x^{2}-2 x-2 x+4\right) \newline Step 3=(4x3)(x24x+4) \stackrel{\text { Step } 3}{\hookrightarrow}=(4 x-3)\left(x^{2}-4 x+4\right) \newline Step 4=4x316x2+16x3x212x12 \stackrel{\text { Step } 4}{\hookrightarrow}=4 x^{3}-16 x^{2}+16 x-3 x^{2}-12 x-12 \newline Step 5=4x319x2+4x12 \stackrel{\text { Step } 5}{\hookrightarrow}=4 x^{3}-19 x^{2}+4 x-12 \newlineFor this reason, Mabel stated that the equation is a true identity.\newlineIs Mabel correct? If not, in which step did she make a mistake?\newlineChoose 11 answer:\newline(A) Mabel is correct.\newline(B) Mabel is incorrect. She made a mistake in step 22.\newline(C) Mabel is incorrect. She made a mistake in step 33.\newline(D) Mabel is incorrect. She made a mistake in step 44.

Full solution

Q. Mabel was asked whether the following equation is an identity:\newline(4x3)(x2)2=4x319x2+4x12 (4 x-3)(x-2)^{2}=4 x^{3}-19 x^{2}+4 x-12 \newlineShe performed the following steps:\newline(4x3)(x2)2 (4 x-3)(x-2)^{2} \newline Step 1=(4x3)(x2)(x2) \stackrel{\text { Step } 1}{\hookrightarrow}=(4 x-3)(x-2)(x-2) \newline Step 2=(4x3)(x22x2x+4) \stackrel{\text { Step } 2}{\hookrightarrow}=(4 x-3)\left(x^{2}-2 x-2 x+4\right) \newline Step 3=(4x3)(x24x+4) \stackrel{\text { Step } 3}{\hookrightarrow}=(4 x-3)\left(x^{2}-4 x+4\right) \newline Step 4=4x316x2+16x3x212x12 \stackrel{\text { Step } 4}{\hookrightarrow}=4 x^{3}-16 x^{2}+16 x-3 x^{2}-12 x-12 \newline Step 5=4x319x2+4x12 \stackrel{\text { Step } 5}{\hookrightarrow}=4 x^{3}-19 x^{2}+4 x-12 \newlineFor this reason, Mabel stated that the equation is a true identity.\newlineIs Mabel correct? If not, in which step did she make a mistake?\newlineChoose 11 answer:\newline(A) Mabel is correct.\newline(B) Mabel is incorrect. She made a mistake in step 22.\newline(C) Mabel is incorrect. She made a mistake in step 33.\newline(D) Mabel is incorrect. She made a mistake in step 44.
  1. Expand Binomial 11: Expand the first binomial.\newline(4x3)(x2)(x2)(4x-3)(x-2)(x-2)
  2. Expand Binomial 22: Expand the second binomial.\newline(4x3)(x22x2x+4)(4x-3)(x^2-2x-2x+4)\newlineOops, made a mistake here. It should be (4x3)(x22x2x+4)=(4x3)(x24x+4)(4x-3)(x^2-2x-2x+4) = (4x-3)(x^2-4x+4)

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