if Z = 2 + i than find the value of Z^2 + Z+ 1

Sure, let's solve this step by step:

1. First, we know that Z = 2 + i.

2. We need to find the value of Z^2. To do this, we square the real and imaginary parts separately:

   (2 + i)^2 = 4 + 4i + i^2.

   But we know that i^2 = -1 (by definition of the imaginary unit), so this simplifies to:

   4 + 4i - 1 = 3 + 4i.

   So, Z^2 = 3 + 4i.

3. Now, we add Z (which is 2 + i) to Z^2:

   Z^2 + Z = (3 + 4i) + (2 + i) = 5 + 5i.

4. Finally, we add 1 to the result:

   Z^2 + Z + 1 = (5 + 5i) + 1 = 6 + 5i.

So, the value of Z^2 + Z + 1 when Z = 2 + i is 6 + 5i.