Once you have something like $x^2 - 4x$ inside parentheses, think about how to rewrite it as a perfect square plus or minus a constant (completing the square).

After completing the square inside the parentheses, remember that the -3 multiplies both terms inside. Distribute it and then combine any constant terms outside the parentheses.

In Desmos, type f(x) = -3x^2 + 12x - 5 to graph the original quadratic.

Type g(x) = -3(x-2)^2 + 7, h(x) = -3(x+2)^2 + 7, p(x) = 3(x-2)^2 + 7, and q(x) = -3(x-2)^2 - 7. Each will appear as a different parabola.

Zoom out or adjust the view so you can clearly see all the parabolas. The correct choice is the one whose graph lies exactly on top of the graph of $f(x)$ for all $x$ (same vertex, same direction of opening, same width). The others will be shifted, flipped, or vertically moved.

Start with the expression:

Factor $-3$ from the $x^2$ and $x$ terms:

Now you are set up to complete the square on $x^2 - 4x$ inside the parentheses.

Look at $x^2 - 4x$.

To complete the square:

• Take half of $-4$, which is $-2$.
• Square it: $(-2)^2 = 4$.

So:

Substitute this back into the expression:

Distribute $-3$ across the two terms inside the parentheses:

So the expression is now written as

This is almost in the same form as the answer choices; you just need to simplify the constants.

Combine the constant terms $12$ and $-5$:

So the original expression is equivalent to

This matches choice A) $-3(x-2)^2 + 7$, so A is the correct answer.