If f(x)=6x^3 - ax^2 + bx + c when a, b and c are constant when f(x) is divided by x^2 - 4 the remainder is 23x - 26. When f(x) divided by (x + 3) the remainder is -220. Find:

(A) the value of a, b and c
(B) the solution of the equation f(x)=0

To find the value of a, b, and c, we need to use the information given about remainders when f(x) is divided by (x^2 - 4) and (x + 3).

Let's start by finding the remainder when f(x) is divided by (x^2 - 4).
The given remainder is 23x - 26.

We know that (x^2 - 4) can be factored as (x - 2)(x + 2).

So, we can set up the following equation:

f(x) = (x^2 - 4) * q(x) + (23x - 26)

Here, q(x) represents the quotient when f(x) is divided by (x^2 - 4).

Now, let's divide f(x) by (x + 3) and set the remainder as -220.

We get the following equation:

f(x) = (x + 3) * r(x) - 220

Here, r(x) represents the quotient when f(x) is divided by (x + 3).

Now, let's substitute f(x) with its expression and solve the equations:

6x^3 - ax^2 + bx + c = (x^2 - 4) * q(x) + (23x - 26) -- equation (1)
6x^3 - ax^2 + bx + c = (x + 3) * r(x) - 220 -- equation (2)

Now, we can simplify equation (1) as:

6x^3 - ax^2 + bx + c = x^2 * q(x) - 4q(x) + 23x - 26

Comparing coefficients of corresponding powers of x, we can conclude the following:

Coefficient of x^3: 6 = 0 (since there is no x^3 term in the remainder)
Coefficient of x^2: -a = 1 (since there is no x^2 term in the remainder)
Coefficient of x: b = 23
Coefficient of constant term: c = - 26 - 4q(x) (since the constant term in the remainder is -26)

We can also simplify equation (2) as:

6x^3 - ax^2 + bx + c = x * r(x) + 3r(x) - 220

Comparing coefficients of corresponding powers of x, we can conclude the following:

Coefficient of x^3: 6 = 0 (since there is no x^3 term in the remainder)
Coefficient of x^2: -a = 0 (since there is no x^2 term in the remainder)
Coefficient of x: b = 1 (since there is no x term in the remainder)
Coefficient of constant term: c = 3r(x) - 220

Now, we have a system of equations:

-a = 1 -- equation (3)
b = 23 -- equation (4)
c = -26 - 4q(x) -- equation (5)
c = 3r(x) - 220 -- equation (6)

Solving equations (3) and (4), we find that:
a = -1
b = 23

Substituting these values into equations (5) and (6), we get:
c = -26 - 4q(x)
c = 3r(x) - 220

Since the values of q(x) and r(x) are not given, we cannot calculate the exact values of a, b, and c without further information.

To find the solutions of the equation f(x) = 0, we can substitute the values of a, b, and c into f(x) and solve for x:

f(x) = 6x^3 - ax^2 + bx + c

Substituting a = -1, b = 23, and c = -26, we have:

f(x) = 6x^3 - (-1)x^2 + 23x + (-26)
f(x) = 6x^3 + x^2 + 23x - 26

Now, set f(x) equal to zero and solve for x:

6x^3 + x^2 + 23x - 26 = 0

To solve this equation, we can use various methods such as factoring, completing the square, or using numerical methods like the Newton-Raphson method. Depending on the specific values of a, b, and c, the solutions of the equation f(x) = 0 can be determined.