Okay, so I need to compute 99 squared plus 99 plus 1 in my head. Hmm, let me think about the best way to approach this. I remember that squaring numbers close to 100 might have a trick or something. Let me try to recall. Maybe there's a formula that can help simplify this calculation. 

Wait, 99 is just 1 less than 100, right? So maybe I can use the identity (a - b)^2, where a is 100 and b is 1. Let me write that out. If 99 is 100 minus 1, then 99 squared would be (100 - 1)^2. Expanding that, it's 100^2 - 2*100*1 + 1^2, which is 10000 - 200 + 1. Let me check that: 100 squared is 10,000, minus 200 is 9,800, plus 1 is 9,801. So 99 squared is 9,801. 

Now, the problem is 99^2 + 99 + 1. So substituting the value I just found, that's 9,801 + 99 + 1. Let me compute that step by step. First, 9,801 + 99. Hmm, adding 100 to 9,801 would be 9,901, but since it's 99, that's one less. So 9,901 minus 1 is 9,900. Then adding the last 1 gives 9,901. Wait, that seems too high. Let me verify again. 

Wait, 99 squared is 9,801. Then adding 99 to that: 9,801 + 99. Let's break it down. 9,801 + 100 is 9,901, but since we're adding 99, subtract 1: 9,901 - 1 = 9,900. Then add the remaining 1: 9,900 + 1 = 9,901. So the total is 9,901. That seems correct. 

Alternatively, maybe there's a smarter way to do this without breaking it into two parts. Let me think. The original expression is 99^2 + 99 + 1. Maybe factor something out? Let's see. 

Wait, 99^2 + 99 is 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. Which is 9900 + 1 = 9901. Oh, that's a much quicker way! So instead of computing 99 squared first, I could factor out the 99. Let me confirm that. 

Yes, 99^2 + 99 = 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. 99*100 is 9900, plus 1 is 9901. That's much simpler. So I should have thought of that first. 

But why didn't I think of that earlier? Maybe because I was focused on expanding 99 squared. But factoring out the common term is a better approach here. 

So, the key insight here is recognizing that 99^2 + 99 can be factored into 99*(99 + 1), which simplifies the calculation. Then adding 1 gives the final answer. 

Let me verify this again step by step to make sure I didn't make a mistake. 

First, 99^2. As I said, 100^2 is 10,000. Then (100 - 1)^2 is 10,000 - 200 + 1 = 9,801. Then 99^2 + 99 is 9,801 + 99. Adding those together: 9,801 + 99. Since 99 is 100 - 1, adding 100 to 9,801 is 9,901, subtract 1 gives 9,900. Then adding 1 gives 9,901. 

Alternatively, factoring: 99^2 + 99 = 99*(99 + 1) = 99*100 = 9,900. Then 9,900 + 1 = 9,901. Both methods lead to the same answer. 

So, the answer is 9,901. 

Wait, let me check once more with a different approach. Maybe compute 99^2 + 99 + 1 directly. 

99^2 is 9,801. Then 9,801 + 99 is 9,900. Then 9,900 + 1 is 9,901. Yep, that's consistent. 

Alternatively, perhaps using another identity. For example, n^2 + n + 1 can sometimes be represented as (n(n + 1)) + 1. Which is exactly what we did here. So, 99*100 + 1. 

Another way to think about it: 99*100 is straightforward, since multiplying by 100 is just adding two zeros. So 99*100 is 9,900. Then adding 1 gives 9,901. 

So, all methods converge to the same answer. Therefore, I'm confident that 99^2 + 99 + 1 equals 9,901. 

Just to recap the steps: 

1. Recognize that 99 is close to 100, so use the square formula (a - b)^2.

2. Alternatively, factor out 99 from the first two terms to simplify the calculation.

3. Either way, the answer comes out to 9,901 after performing the arithmetic step by step. 

Therefore, the final answer is 9,901.
Therefore, the result of $99^2 + 99 + 1$ is $\boxed{9901}$.
</think>