Sure, let's go through the steps and provide the justifications.
For Step A:
[tex]\[
a\left(x+\frac{b}{2a}\right)^2 = -c + \frac{b^2}{4a}
\][/tex]
[tex]\[
a\left(x+\frac{b}{2a}\right)^2 = \frac{-4ac + b^2}{4a}
\][/tex]
Justification: We rewrote [tex]\(-c\)[/tex] with a common denominator to match the denominator from the second term in the equation. This makes it possible to combine these terms into a single fraction:
[tex]\[
\frac{-4ac}{4a} + \frac{b^2}{4a} = \frac{-4ac + b^2}{4a}
\][/tex]
Here, the justification is indeed the "common denominator."
For Step B:
[tex]\[
a\left(x+\frac{b}{2a}\right)^2 = \frac{-4ac + b^2}{4a}
\][/tex]
[tex]\[
\left(\frac{1}{a}\right) a \left(x+\frac{b}{2a}\right)^2 = \left(\frac{1}{a}\right) \left(\frac{-4ac + b^2}{4a}\right)
\][/tex]
Justification: We applied the multiplication property of equality to isolate [tex]\(\left(x+\frac{b}{2a}\right)^2\)[/tex]. Multiplying both sides of the equation by [tex]\(\frac{1}{a}\)[/tex] ensures that the equation remains balanced.
Thus, for Step B, the correct justification is the "multiplication property of equality."
