Sure, let's solve the expression [tex]\((x-2)^2 + (y+2)^2\)[/tex] step by step.
### Step 1: Expand [tex]\((x-2)^2\)[/tex]
First, recall that the square of a binomial is given by:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Applying this formula to [tex]\((x-2)^2\)[/tex]:
[tex]\[
(x-2)^2 = x^2 - 2 \cdot x \cdot 2 + 2^2 = x^2 - 4x + 4
\][/tex]
### Step 2: Expand [tex]\((y+2)^2\)[/tex]
Similarly, recall that:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Applying this formula to [tex]\((y+2)^2\)[/tex]:
[tex]\[
(y+2)^2 = y^2 + 2 \cdot y \cdot 2 + 2^2 = y^2 + 4y + 4
\][/tex]
### Step 3: Combine the expanded expressions
Now we need to add the two expanded expressions together:
[tex]\[
(x-2)^2 + (y+2)^2 = (x^2 - 4x + 4) + (y^2 + 4y + 4)
\][/tex]
### Step 4: Simplify the result
Combine like terms:
[tex]\[
x^2 - 4x + 4 + y^2 + 4y + 4
\][/tex]
This simplifies to:
[tex]\[
x^2 + y^2 - 4x + 4y + 8
\][/tex]
### Final Answer
Thus, the simplified form of the expression [tex]\((x-2)^2 + (y+2)^2\)[/tex] is:
[tex]\[
\boxed{x^2 + y^2 - 4x + 4y + 8}
\][/tex]
