Given:
p(x) = ax^3 + bx^2 + x - 6
p(x) leaves a remainder of 4 when divided by (x - 2)

To prove:
a = 0 and b = 2

Proof:

1. Division Algorithm:
The Division Algorithm states that any polynomial p(x) can be divided by a linear polynomial (x - c), resulting in a quotient q(x) and a remainder r. Mathematically, it can be represented as:
p(x) = (x - c) * q(x) + r
In this case, p(x) can be divided by (x - 2), leaving a remainder of 4.

2. Dividing p(x) by (x - 2):
Using the Division Algorithm, we can divide p(x) by (x - 2) and find the remainder. Let's perform the division:

__________________________
(x - 2) | ax^3 + bx^2 + x - 6
- (ax^2 - 2ax^2)
__________________________
(bx^2 + 2ax^2 + x)
- (bx - 2bx)
__________________________
(2ax^2 + bx + x)
- (2ax - 4ax)
__________________________
(bx + 5ax - 6)
- (bx - 2bx)
__________________________
(5ax + 2bx - 6)
- (5ax - 10ax)
__________________________
(2bx + 5ax - 6)
- (2bx - 4bx)
__________________________
(5bx + 5ax - 6)
- (5bx - 10bx)
__________________________
(5ax + 5bx - 6)
- (5ax - 10ax)
__________________________
(5bx - 5ax - 6)
- (5bx - 10bx)
__________________________
(-5ax - 6)

The remainder obtained after the division is -5ax - 6.

3. Equating the remainder:
According to the given information, when p(x) is divided by (x - 2), the remainder is 4. Therefore, we can equate the remainder we obtained with 4:

-5ax - 6 = 4

Simplifying the equation, we have:

-5ax = 10

Dividing both sides of the equation by -5a, we get:

x = -2

4. Substituting x = -2 in p(x):
Substituting x = -2 in p(x) = ax^3 + bx^2 + x - 6, we have:

p(-2) = a(-2)^3 + b(-2)^2 + (-2) - 6
p(-2) = -8a + 4b - 2 - 6
p(-2) = -8a + 4b - 8

Since p(-2) gives the remainder when divided by (x - 2), which is