for what value of b will be polynomial p(x) = -2x^3 + bx^2 - 5x + 2 have the same remainder when it is divided by x- 2 and by x+1 ?

Divide it by x-2 and get:

-2x^{2}+bx-4x+2b-13+\underbrace{\frac{4b-24}{x-2}}_{\text{remainder}}

Divide by x+1:

-2x^{2}+bx+2x-b-7+\underbrace{\frac{b+9}{x+1}}_{\text{remainder}}

Can you finish?

Look at the remainder terms. Equate the numerators in the remainder terms and solve for b.

noo. Don't you have to sub 1 into 2 to get the other value.

No.

4b-24=b+9

If you use the remainder theorem, then you apply it. It says that for a given polynomial f(x), the remainder, when f(x) is divided by (x - a), is given by f(a).

Hence;

p(x) = -2x^3 + bx^2 - 5x + 2

Substituting 2;

p(2) = -2(2)^3 + b(2)^2 - 5(2) + 2 = -16 + 4b - 10 + 2 = 4b - 24

Substituting -1;

p(-1) = -2(-1)^3 + b(-1)^2 - 5(-1) + 2 = 2 + b + 5 + 2 = 9 + b

If both remainder are equal, then:

4b - 24= 9 + b

Solve for b.