The quadratic $-6x^2+36x+216$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?
We complete the square.

Factoring $-6$ out of the quadratic and linear terms gives $-6x^2 + 36x = -6(x^2-6x)$.

Since $(x-3)^2 = x^2 - 6x + 9$, we can write $$-6(x-3)^2 = -6x^2 + 36x - 54.$$This quadratic agrees with the given $-6x^2 + 36x + 216$ in all but the constant term. We can write

\begin{align*}
-6x^2 + 36x + 216 &= (-6x^2 + 36x - 54) + 270 \\
&= -6(x-3)^2 + 270.
\end{align*}Therefore, $a=-6$, $b=-3$, $c=270$, and $a+b+c = -6-3+270 = \boxed{261}$.
261
Level 5
Algebra
Problem: The quadratic $-6x^2+36x+216$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To convert the quadratic expression $-6x^2 + 36x + 216$ into the form $a(x+b)^2+c$, we first identify the coefficients of the quadratic. The coefficient of $x^2$ is $-6$, which indicates that $a = -6$. 

Next, we need to complete the square for the quadratic expression. We start with the expression $-6(x^2 - 6x) + 216$. To complete the square inside the parentheses, we take the coefficient of $x$, which is $-6$, halve it to get $-3$, and then square it to obtain $9$. 

We rewrite the expression as follows: 
$-6(x^2 - 6x + 9 - 9) + 216$. 
This simplifies to:
$-6((x - 3)^2 - 9) + 216$. 

Distributing $-6$ gives:
$-6(x - 3)^2 + 54 + 216$. 

Combining the constants results in:
$-6(x - 3)^2 + 270$. 

Now we have the expression in the desired form $a(x+b)^2+c$, where $a = -6$, $b = -3$, and $c = 270$. 

To find $a + b + c$, we calculate:
$-6 + (-3) + 270 = -6 - 3 + 270 = 261$. 

Thus, the final answer is $261$.<|END_OF_LATENT|> The final answer is $261$.