(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

Let's find the sum of the last two expressions:

We have:

x + 2x² - 3x³ and 2 - x² + 6x - x³

Arranging the terms in descending powers of x and use 0 as a placeholder for any missing term:

$\begin{array}{rcccccc} -3x^3 & + & 2x^2 & + & x & + & 0 \\ -\phantom{3} x^3 & - & x^2 & + & 6x & + & 2 \\ \hline -4x^3 & + & x^2 & + & 7x & + & 2 \\ \hline \end{array}$

The sum is -4x³ + x² + 7x + 2

Now, subtract 6x³ - 5x² + 4x - 3 from the above sum:

$\begin{array}{rcccccc} -4x^3 & + & x^2 & + & 7x & + & 2 \\ +6x^3 & - & 5x^2 & + & 4x & - & 3 \\ -\phantom{6x^3} & + & & - & & + \\ \hline -10x^3 & + & 6x^2 & + & 3x & + & 5 \\ \hline \end{array}$

Hence, the answer is -10x³ + 6x² + 3x + 5

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

Let's find the sum of first pair:

We have:

a + 2b - 3c and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 2b & - & 3c \\ -4a & - & 3b & + & 2c \\ \hline -3a & - & b & - & c \\ \hline \end{array}$

Sum 1 = -3a - b - c

Now, let's find the sum of second pair:

We have:

5b - 4c + a and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 5b & - & 4c \\ -4a & - & 3b & + & 2c \\ \hline -3a & + & 2b & - & 2c \\ \hline \end{array}$

Sum 2 = -3a + 2b - 2c

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} -3a & + & 2b & - & 2c \\ -3a & - & b & - & c \\ +\phantom{3a} & + & & + \\ \hline 0 & + & 3b & - & c \\ \hline \end{array}$

Hence, the answer is 3b - c

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

Let's find the sum of first pair:

We have:

x² - 5xy + 2y² and y² - 2xy - 3x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} x^2 & - & 5xy & + & 2y^2 \\ -3x^2 & - & 2xy & + & y^2 \\ \hline -2x^2 & - & 7xy & + & 3y^2 \\ \hline \end{array}$

Sum 1 = -2x² - 7xy + 3y²

Now, let's find the sum of second pair:

We have:

6x² - 8xy - y² and 2xy - 2y² - x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} 6x^2 & - & 8xy & - & y^2 \\ -x^2 & + & 2xy & - & 2y^2 \\ \hline 5x^2 & - & 6xy & - & 3y^2 \\ \hline \end{array}$

Sum 2 = 5x² - 6xy - 3y²

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} 5x^2 & - & 6xy & - & 3y^2 \\ -2x^2 & - & 7xy & + & 3y^2 \\ +\phantom{2x^2} & + & & - \\ \hline 7x^2 & + & xy & - & 6y^2 \\ \hline \end{array}$

Hence, the answer is 7x² + xy - 6y²