To find the value of 'a' when the polynomials (ax^3 + 3x^2 - 3) and (2x^3 - 5x + a) leave the same remainder when divided by (x - 4), we can apply the remainder theorem. The remainder theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is equal to f(c).

1. Apply the remainder theorem to the first polynomial:
When the polynomial (ax^3 + 3x^2 - 3) is divided by (x - 4), the remainder is equal to f(4), where f(x) = ax^3 + 3x^2 - 3.
Let's calculate the remainder using f(4).
f(4) = a(4)^3 + 3(4)^2 - 3
= 64a + 48 - 3
= 64a + 45

2. Apply the remainder theorem to the second polynomial:
When the polynomial (2x^3 - 5x + a) is divided by (x - 4), the remainder is equal to f(4), where f(x) = 2x^3 - 5x + a.
Let's calculate the remainder using f(4).
f(4) = 2(4)^3 - 5(4) + a
= 2(64) - 20 + a
= 128 - 20 + a
= 108 + a

3. Set the remainders equal to each other:
Since both polynomials leave the same remainder when divided by (x - 4), we can set the remainders equal to each other and solve for 'a'.
64a + 45 = 108 + a

4. Solve for 'a':
To find the value of 'a', we need to solve the equation:
64a - a = 108 - 45
63a = 63
a = 1

Therefore, the value of 'a' is 1.

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