When the expression $-2x^2-20x-53$ is written in the form $a(x+d)^2+e$, where $a$, $d$, and $e$ are constants, then what is the sum $a+d+e$?

We start with  and wish to write it as 

First, pull 2 out from the first two terms: 

Let’s look at what is in parenthesis. In the final form this needs to be a perfect square. Right now we have  and we can obtain -10x by adding -5x and -5x. That is, we can build the following perfect square: 

The “problem” with what we just did is that we added to what was given. Let’s put the expression together. We have  and when we multiply that out it does not give us what we started with. It gives us 

So you see our expression is not right. It should have a -53 but instead has a -3. So to correct it we need to subtract another 50.

We do this as follows:  which gives us the final expression we seek:



If you multiply this out you will get the exact expression we were given. This means that:
a = 2
d = -5
e =  -103

We are asked for the sum of a, d and e which is 2 + (-5) + (-103) = -106

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