Write $P(x) = ax^2 + bx + c$ and then find $P(1)$ in terms of $a$, $b$, and $c$. How is that related to $a + b + c$?

Once you know $a + b + c = P(1)$, substitute $x = 1$ directly into $(3x - 2)^3 - (3x - 2)(9x^2 - 4) + 8$ and simplify step by step.

After substituting $x = 1$, simplify the parentheses first, then the exponents and products, and finally add and subtract carefully.

Remember that if the expression equals $ax^2 + bx + c$, then $a + b + c$ is the value of the expression at $x = 1$. So you want to evaluate the given expression when $x = 1$.

In Desmos, type the expression with 1 in place of $x$:

(3*1 - 2)^3 - (3*1 - 2)*(9*1^2 - 4) + 8

Desmos will display a single number as the output; that number is the value of the expression at $x = 1$, which equals $a + b + c$.

Let

We are told that this can be written as $P(x) = ax^2 + bx + c$.

Now evaluate $P(1)$:

So instead of expanding everything, we just need to find the value of the original expression when $x=1$.

Plug $x = 1$ into each place you see $x$:

Now simplify inside the parentheses:

• $3(1) - 2 = 3 - 2 = 1$
• $9(1)^2 - 4 = 9 - 4 = 5$

So the expression becomes

Now evaluate the power and the multiplication:

• $1^3 = 1$
• $1 \times 5 = 5$

So the expression simplifies to

Compute the remaining arithmetic:

This value is $P(1)$, which we found equals $a + b + c$. Therefore,