Dưới đây là các bước rút gọn của từng biểu thức:

### 1) \( x(x+2)+(x+1)(x-3) \)
- Rút gọn:
\[
x^2 + 2x + (x^2 - 3x + x - 3) = 2x^2 - 3
\]

### 2) \( (x+2)(x^2-2x+4) - x^2(x+1) \)
- Rút gọn:
\[
(x^3 - 2x^2 + 4x + 2x^2 - 4x + 8) - (x^3 + x^2) = 2x^2 + 8
\]

### 3) \( (x+2)(2x-5) - (2x+1)(x-3) \)
- Rút gọn:
\[
(2x^2 - 5x + 4x - 10) - (2x^2 - 6x + x - 3) = -2x - 7
\]

### 4) \( (x+y)^2 - 2(x-y)(x+y) - (x-y)^2 \)
- Rút gọn:
\[
(x^2 + 2xy + y^2) - 2(x^2 - y^2) - (x^2 - 2xy + y^2) = 4xy
\]

### 5) \( 2x^2(x-3) + 4(2x^2 - x) \)
- Rút gọn:
\[
2x^3 - 6x^2 + 8x^2 - 4x = 2x^3 + 2x^2 - 4x
\]

### 6) \( (2x-1)(2x+1) - (x-2)^2 \)
- Rút gọn:
\[
(4x^2 - 1) - (x^2 - 4x + 4) = 3x^2 + 4x - 5
\]

### 7) \( x(x-2) + (x-1)(x+1) + 2x + 1 \)
- Rút gọn:
\[
x^2 - 2x + (x^2 - 1) + 2x + 1 = 2x^2
\]

### 8) \( (x-y)^2 + (x+y)^2 + 2(x-y)(x+y) \)
- Rút gọn:
\[
x^2 - 2xy + y^2 + x^2 + 2xy + y^2 + 2(x^2 - y^2) = 4x^2
\]

### 9) \( (x-3)(x+3) - (x-1)(x+1) \)
- Rút gọn:
\[
x^2 - 9 - (x^2 - 1) = -8
\]

### 10) \( (x+2)(x^2-2x+4) - (45 + x^3) \)
- Rút gọn:
\[
x^3 - 2x^2 + 4x + 2x^2 - 4x + 8 - 45 - x^3 = -37
\]

### 11) \( 5x(2x^2-3x+1) \)
- Rút gọn:
\[
10x^3 - 15x^2 + 5x
\]

### 12) \( (x+3)(x^2-3x+2) \)
- Rút gọn:
\[
x^3 - 3x^2 + 2x + 3x^2 - 9x + 6 = x^3 - 7x + 6
\]

### 13) \( (x+5)(x-5) - x^2 + 7x - 2 \)
- Rút gọn:
\[
x^2 - 25 - x^2 + 7x - 2 = 7x - 27
\]

### 14) \( (x-2)^2 - x(x+3) \)
- Rút gọn:
\[
x^2 - 4x + 4 - x^2 - 3x = -7x + 4
\]

### 15) \( 3x(2x^2 - 7x + 4) \)
- Rút gọn:
\[
6x^3 - 21x^2 + 12x
\]

### 16) \( 3x^2 - 2x(x+5) + 7 \)
- Rút gọn:
\[
3x^2 - 2x^2 - 10x + 7 = x^2 - 10x + 7
\]

### 17) \( (x-3)(x+3) - (x+2)(x-1) \)
- Rút gọn:
\[
x^2 - 9 - (x^2 - x - 2) = -7 + x
\]

### 18) \( (x+1)^2 + (2x-1)^2 + (x+1)(4x-2) \)
- Rút gọn:
\[
x^2 + 2x + 1 + 4x^2 - 4x + 1 + 4x^2 + 2x - 2 = 9x^2 + 4x
\]

### 19) \( 3x(x+2) - 2(3x^2 - 1) \)
- Rút gọn:
\[
3x^2 + 6x - 6x^2 + 2 = -3x^2 + 6x + 2
\]

### 20) \( (2x-1)^2 - (x+7)(x-7) \)
- Rút gọn:
\[
4x^2 - 4x + 1 - (x^2 - 49) = 3x^2 - 4x + 50
\]

### 21) \( 2xy(3x^2y - \frac{1}{2}x) \)
- Rút gọn:
\[
6x^3y^2 - x^2y
\]

### 22) \( (2x-1)(x^2 + 4x - 3) \)
- Rút gọn:
\[
2x^3 + 8x^2 - 6x - x^2 - 4x + 3 = 2x^3 + 7x^2 - 10x + 3
\]

### 23) \( (x+2)^2 - (x-3)(x+1) \)
- Rút gọn:
\[
x^2 + 4x + 4 - (x^2 - 2x - 3) = 6x + 7
\]

### 24) \( (2x-1)(2x+1) - (x+1)^2 + 2x \)
- Rút gọn:
\[
4x^2 - 1 - (x^2 + 2x + 1) + 2x = 3x^2
\]

### 25) \( 2x(x+2) - x(2x+1) \)
- Rút gọn:
\[
2x^2 + 4x - 2x^2 - x = 3x
\]

### 26) \( 4(x-1)(x+1) - (2x-1)^2 \)
- Rút gọn:
\[
4(x^2 - 1) - (4x^2 - 4x + 1) = 0x^2 + 4x - 5
\]

### 27) \( (x-2)^2 - (x-3)(x+3) \)
- Rút gọn:
\[
x^2 - 4x + 4 - (x^2 - 9) = -4x + 13
\]

### 28) \( 2x^2(x-3) + 4(2x^2 - 1) \)
- Rút gọn:
\[
2x^3 - 6x^2 + 8x^2 - 4 = 2x^3 + 2x^2 - 4
\]

### 29) \( (2x-1)(2x+1) - (x-2)^2 \)
- Rút gọn:
\[
4x^2 - 1 - (x^2 - 4x + 4) = 3x^2 + 4x - 5
\]

### 30) \( 3x^2 - x(2x+1) \)
- Rút gọn:
\[
3x^2 - 2x^2 - x = x^2 - x
\]

### 31) \( (x-2)^2 + (x+3)(x-3) - 2x(x+5) \)
- Rút gọn:
\[
(x^2 - 4x + 4) + (x^2 - 9) - (2x^2 + 10x) = -x^2 - 14x - 5
\]

### 32) \( 2x^3y(x^2 - 2y + 1) \)
- Rút gọn:
\[
2x^5y - 4x^3y^2 + 2x^3y
\]

### 33) \( (x+y)^2 + (x-y)^2 - 2(x+y)(x-y) \)
- Rút gọn:
\[
(x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) - 2(x^2 - y^2) = 4y^2
\]

### 34) \( 5x^2 - 3x(x+2) \)
- Rút gọn:
\[
5x^2 - 3x^2 - 6x = 2x^2 - 6x
\]

### 35) \( (x+4)^2 - (x+5)(x-5) - 2x(x+1) \)
- Rút gọn:
\[
x^2 + 8x + 16 - (x^2 - 25) - (2x^2 + 2x) = -2x^2 + 6x + 41
\]

### 36) \( 2(x+3) - x^2 - 3x \)
- Rút gọn:
\[
2x + 6 - x^2 - 3x = -x^2 - x + 6
\]

### 37) \( (2x+1)^2 - (4x-1)(x-3) - 15 \)
- Rút gọn:
\[
4x^2 + 4x + 1 - (4x^2 - 13x + 3) - 15 = 17x - 17
\]

### 1) x(x + 2) + (x + 1)(x - 3)
\[
= x^2 + 2x + (x^2 - 3x + x - 3)
= x^2 + 2x + x^2 - 2x - 3
= 2x^2 - 3
\]

### 2) (x + 2)(x^2 - 2x + 4) - x^2(x + 1)
\[
= (x^3 - 2x^2 + 4x + 2x^2 - 4x + 8) - (x^3 + x^2)
= x^3 + 8 - x^3 - x^2
= 8 - x^2
\]

### 3) (x + 2)(2x - 5) - (2x + 1)(x - 3)
\[
= (2x^2 - 5x + 4x - 10) - (2x^2 - 6x + x - 3)
= (2x^2 - x - 10) - (2x^2 - 5x - 3)
= 2x^2 - x - 10 - 2x^2 + 5x + 3
= 4x - 7
\]

### 4) (x + y)^2 - 2(x - y)(x + y) - (x - y)^2
\[
= (x^2 + 2xy + y^2) - 2(x^2 - y^2) - (x^2 - 2xy + y^2)
= x^2 + 2xy + y^2 - 2x^2 + 2y^2 - x^2 + 2xy - y^2
= 4xy
\]

### 5) 2x^2(x - 3) + 4(2x^2 - x)
\[
= 2x^3 - 6x^2 + 8x^2 - 4x
= 2x^3 + 2x^2 - 4x
\]

### 6) (2x - 1)(2x + 1) - (x - 2)^2
\[
= 4x^2 - 1 - (x^2 - 4x + 4)
= 4x^2 - 1 - x^2 + 4x - 4
= 3x^2 + 4x - 5
\]

### 7) x(x - 2) + (x - 1)(x + 1) + 2x + 1
\[
= x^2 - 2x + (x^2 - 1) + 2x + 1
= x^2 - 2x + x^2 - 1 + 2x + 1
= 2x^2
\]

### 8) (x - y)^2 + (x + y)^2 + 2(x - y)(x + y)
\[
= (x^2 - 2xy + y^2) + (x^2 + 2xy + y^2) + 2(x^2 - y^2)
= x^2 - 2xy + y^2 + x^2 + 2xy + y^2 + 2x^2 - 2y^2
= 4x^2
\]

### 9) (x - 3)(x + 3) - (x - 1)(x + 1)
\[
= x^2 - 9 - (x^2 - 1)
= x^2 - 9 - x^2 + 1
= -8
\]

### 10) (x + 2)(x^2 - 2x + 4) - (45 + x^3)
\[
= x^3 + 8 - (45 + x^3)
= 8 - 45
= -37
\]