The remainder is 44
Explanations:The given polynomial is:
[tex]4x^4-3x^3+2x^2-5x\text{ + 6}[/tex]The function is to be divided by x - 2
The remainder of the division will be calculated using the remainder theorem
The remainder theorem states that " If a function g(x) is divided by x - a, the remainder of the division will be g(a)"
[tex]\text{Let g(x) = 4x}^4-3x^3+2x^2-5x+6[/tex]The remainder when g(x) is divided by x -2 is g(2)
[tex]\begin{gathered} g(2)=4(2)^4-3(2)^3+2(2)^2-\text{ 5(2) + 6} \\ g(2)\text{ = 4(16) - 3(8) + 2(4) -5(2) + 6} \\ g(2)\text{ = 64 - 24 + 8 - 10 + 6} \\ g(2)\text{ = 44} \end{gathered}[/tex]