What is the polynomial with highest degree 3 such that when the polynomial is divided by binomials x+2 and x^2-1, the remainder is 3.

The polynomial is not determined, yet. It will be
determined, when all it's coefficients will be known.

We'll
write the polynomial whose leading term is ax^3 as:

P =
ax^3 + bx^2 + cx + d

Since we know the reminder that we've
get when P is divided by given binomials, we'll apply the reminder
theorem.

We notice that x^2 - 1 is a difference of 2
squares and it represents the product:

x^2 - 1 =
(x-1)(x+1)

We'll write the reminder theorem, when P(x) is
divided by (X-1):

P(1)=3

We'll
write the reminder theorem, when P(x) is divided by
(X+1):

P(-1)=3

We'll write the
reminder theorem, when P(x) is divided by
(X+2):

P(-2)=3

From these
facts, we notice that the reminder of the division of   P(x) to the product of
polynomials (X-1)(X+1)(X+2) is also 3.

We'll write the
reminder theorem:

aX^3+bX^2+cX+d=(X-1)(X+1)(X+2) +
3

We'll remove the
brackets:

aX^3+bX^2+cX+d=(X^2-1)(X+2)
+3

aX^3+bX^2+cX+d = X^3 + 2X^2-X -2 +
3

We'll combine like terms and we'll
get:

aX^3 + bX^2 + cX + d = X^3 + 2X^2 - X +
1

The requested polynomial is: P(X) = X^3 +
2X^2 - X + 1.