None of these questions have anything to do directly with the polynomial remainder theorem. The theorem says that the remainder upon dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] is given by the value of [tex]f(c)[/tex].
For these questions, all you really have to do is evaluate the given polynomials at the given points, and IMO is much less work.
Question 2: [tex]f(5)=4(5)^2-5(5)+2=77[/tex]
Question 3: [tex]f(4)=3(4)^4-8)4)^2-2(4)+12=644[/tex]
Question 4: Here you have check the value of [tex]h(x)[/tex] and 2 and -2, then interpret them as points in the coordinate plane, [tex](x,h(x))[/tex].
[tex]h(-2)=2(-2)^5-4(-2)^4-2(-2)^2+15=-121[/tex]
[tex]h(2)=2(2)^5-4(2)^4-2(2)^2+15=7[/tex]
Question 5: Same as in question 4, but you have to check [tex]h(x)[/tex] at -4, -3, -2, -1.
[tex]h(-4)=72[/tex]
[tex]h(-3)=46[/tex]
[tex]h(-2)=26[/tex]
[tex]h(-1)=12[/tex]
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If you insist on using the polynomial remainder theorem, it's a question of polynomial division. For instance, in question 2 you'd compute
[tex]\dfrac{4x^2-5x+2}{x-5}=4x+15+\dfrac{77}{x-5}\implies4x^2-5x+2=(4x+15)(x-5)+77[/tex]
so the remainder is 77, as we found by simply computing [tex]f(5)[/tex].
