After factoring out the greatest common factor, look at the trinomial in parentheses. Does it match the pattern $(a - b)^2 = a^2 - 2ab + b^2$ for some choices of $a$ and $b$?

Once you have written the expression as 3 times a squared binomial in $q$ and $r$, compare that binomial to $p = 2q - r$. How are they related?

In Desmos, type p = 2q - r. Desmos will create sliders for $q$ and $r$ and automatically compute $p$ from them.

On a new line, enter the original expression as something like A = 3r^2 - 12qr + 12q^2. This will give you a numeric value for A for each pair of $q$ and $r$ values.

Create four more expressions, one for each answer choice (for example, label them B, C, D, and E). Use the same variables $p$, $q$, and $r$ that you already defined. Move the sliders for $q$ and $r$ to different values and compare each choice’s value to A; the choice whose value always matches A is the equivalent expression.

Start with the expression:

All three terms share a common factor of 3, so factor out 3:

Now focus on factoring the trinomial in parentheses.

Look at the expression inside the parentheses:

Compare it to the pattern $(a - b)^2 = a^2 - 2ab + b^2$.

Here, $a = r$ and $b = 2q$:

• $a^2 = r^2$
• $2ab = 2 \cdot r \cdot 2q = 4rq$ (which matches the middle term, with a minus sign)
• $b^2 = (2q)^2 = 4q^2$

So,

Our expression becomes:

We currently have $3(r - 2q)^2$. Notice that

When you square a negative, the sign disappears:

So the expression can also be written as:

You are given that

From the previous step, we rewrote the expression as $3(2q - r)^2$. Since $2q - r$ equals $p$, substitute $p$ for $2q - r$:

Therefore, the expression $3r^2 - 12qr + 12q^2$ is equivalent to $3p^2$.