It seems you have provided several algebraic expressions or inequalities. I'll help you simplify or analyze them. However, your message contains some formatting issues. I'll assume you meant to outline different expressions or inequalities, so I'll separate each part:

1. **Expression**: \((x+2)^2 + 2(x+1)\)

**Simplification**:
\[
= (x^2 + 4x + 4) + 2(x + 1)
= x^2 + 4x + 4 + 2x + 2
= x^2 + 6x + 6
\]

2. **Expression**: \((x-4)^2 + (x+1)(x+2)\)

**Simplification**:
\[
= (x^2 - 8x + 16) + (x^2 + 3x + 2)
= x^2 - 8x + 16 + x^2 + 3x + 2
= 2x^2 - 5x + 18
\]

3. **Expression**: \((x-y)^2 + 2x(x+y)\)

**Simplification**:
\[
= (x^2 - 2xy + y^2) + (2x^2 + 2xy)
= x^2 - 2xy + y^2 + 2x^2 + 2xy
= 3x^2 + y^2
\]

4. **Expression**: \(3(x+1) + (x+1)^2\)

**Simplification**:
\[
= 3x + 3 + (x^2 + 2x + 1)
= x^2 + 5x + 4
\]

5. **Expression**: \((x+5)2x - (x-2)^2\)

**Simplification**:
\[
= (2x^2 + 10x) - (x^2 - 4x + 4)
= 2x^2 + 10x - x^2 + 4x - 4
= x^2 + 14x - 4
\]

These are the simplified forms of the expressions you've provided. If you have any specific questions or need further assistance on these expressions or related topics, please let me know!