Answer:

Therefore, the value of k is -29/8.

Step-by-step explanation:

If the polynomial gives the same remainder when divided by (x+1) or (x-3), it means that these two factors are factors of the polynomial.

Therefore, we can set up two equations as follows:

x^3 + kx^2 + x + 10 = (x+1) (ax^2 + bx + c) + d

x^3 + kx^2 + x + 10 = (x-3) (ex^2 + fx + g) + h

where a, b, c, e, f, and g are constants and d and h are the remainders.

We can simplify these equations by substituting x = -1 and x = 3, respectively:

-1+k-1+10=d, which gives d=k+8

3^3+9k+3+10=h, which gives h=9k+37

Since the remainders are the same, we can set d=h and solve for k:

k+8=9k+37

8k= -29

k = -29/8