The expression ax^4 + 4x^3 - bx^2 - 3x + 12 leaves a remainder of 7 when divided by x-2. Find the remainder when divided by x+2.

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This can be solved using the remainder theorem which states that when a function f(x) is divided by (x - a), the remainder is f(a)

Here f(x) = ax^4 + 4x^3 - bx^2 - 3x + 12 and it leaves a remainder of 7 when divided by x - 2

This gives f(2) = a*2^4 + 4*2^3 - b*2^2 - 3*2 + 12 = 7

=> 16a + 32 - 4b - 6 + 12 = 7

=> 16a - 4b = -31

When f(x) is divided by x + 2 the remainder is f(-2)

f(-2) = a*(-2)^4 + 4*(-2)^3 - b*(-2)^2 - 3(-2) + 12

=> 16a - 32 - 4b + 6 + 12

as 16a - 4b = -31

=> -31 - 32 + 6 + 12

=> -45

**The remainder when the given expression is divided by x + 2 is -45**

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