Let's simplify each expression step by step.

### 1. \(-4x^3 (x^3 - 4x^2 + 7x - 3)\)

Distributing \(-4x^3\):

\[
-4x^3 \cdot x^3 + -4x^3 \cdot (-4x^2) + -4x^3 \cdot 7x + -4x^3 \cdot (-3)
\]
\[
= -4x^6 + 16x^5 - 28x^4 + 12x^3
\]

### 2. \(3x - 2x + 5x^2 - x + 1\)

Combining like terms:

\[
(3x - 2x - x) + 5x^2 + 1 = 0x + 5x^2 + 1 = 5x^2 + 1
\]

### 3. \(3x^2(-2x^2 - 5x^2 - x)\)

Distributing \(3x^2\):

\[
3x^2 \cdot (-2x^2) + 3x^2 \cdot (-5x^2) + 3x^2 \cdot (-x)
\]
\[
= -6x^4 - 15x^4 - 3x^3 = -21x^4 - 3x^3
\]

### 4. \(-5x^2y (3x^2y^3 - 2x^3y^2 - xy)\)

Distributing \(-5x^2y\):

\[
-5x^2y \cdot 3x^2y^3 + -5x^2y \cdot (-2x^3y^2) + -5x^2y \cdot (-xy)
\]
\[
= -15x^4y^4 + 10x^5y^3 + 5x^3y^2
\]

### 5. \(4x^3y (-2xy + 4x - 3y^2)\)

Distributing \(4x^3y\):

\[
4x^3y \cdot (-2xy) + 4x^3y \cdot 4x + 4x^3y \cdot (-3y^2)
\]
\[
= -8x^4y^2 + 16x^4y - 12x^3y^3
\]

### 6. \(\frac{1}{2} x^2y (2x^3y^3 - 4xy - 6)\)

Distributing \(\frac{1}{2} x^2y\):

\[
\frac{1}{2} x^2y \cdot 2x^3y^3 + \frac{1}{2} x^2y \cdot (-4xy) + \frac{1}{2} x^2y \cdot (-6)
\]
\[
= x^5y^4 - 2x^3y^2 - 3x^2y
\]

### Summary of Results:

1. \(-4x^6 + 16x^5 - 28x^4 + 12x^3\)
2. \(5x^2 + 1\)
3. \(-21x^4 - 3x^3\)
4. \(-15x^4y^4 + 10x^5y^3 + 5x^3y^2\)
5. \(-8x^4y^2 + 16x^4y - 12x^3y^3\)
6. \(x^5y^4 - 2x^3y^2 - 3x^2y\)

If you need further assistance or additional calculations, feel free to ask!