To determine which expressions can divide both polynomials \( x^3 + 2x^2 - 5x + 2 \) and \( x^3 + 4x^2 + x - 6 \) exactly, we need to check each expression one-by-one:

Step 1: Check the divisibility by \( x-1 \)

We can determine if \( x-1 \) is a factor of a polynomial by substituting \( x = 1 \) into the polynomial and checking if it equals zero.

Substituting \( x = 1 \) into the first polynomial:

\(x^3 + 2x^2 - 5x + 2 = 1^3 + 2(1)^2 - 5(1) + 2 = 1 + 2 - 5 + 2 = 0\)

The first polynomial is divisible by \( x-1 \).

Substituting \( x = 1 \) into the second polynomial:

\(x^3 + 4x^2 + x - 6 = 1^3 + 4(1)^2 + 1 - 6 = 1 + 4 + 1 - 6 = 0\)

The second polynomial is also divisible by \( x-1 \).

Therefore, \( x-1 \) is a common factor.

Step 2: Check the divisibility by \( x+1 \)

Substitute \( x = -1 \) into the first polynomial:

\(x^3 + 2x^2 - 5x + 2 = (-1)^3 + 2(-1)^2 - 5(-1) + 2 = -1 + 2 + 5 + 2 = 8 \neq 0\)

The first polynomial is not divisible by \( x+1 \).

Since \( x+1 \) is not a factor of the first polynomial, it cannot be a common factor.

Step 3: Check the divisibility by \( x+2 \)

Substitute \( x = -2 \) into the first polynomial:

\(x^3 + 2x^2 - 5x + 2 = (-2)^3 + 2(-2)^2 - 5(-2) + 2 = -8 + 8 + 10 + 2 = 12 \neq 0\)

The first polynomial is not divisible by \( x+2 \).

Since \( x+2 \) is not a factor of the first polynomial, it cannot be a common factor.

Conclusion:

The only expression that divides both polynomials exactly is \( x-1 \), which corresponds to the correct answer: I only.