Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]
1. **Assume the degrees of the polynomials:**
   Let \( \deg(f) = m \) and \( \deg(g) = n \). Then, the degree of the left-hand side \( x^2 \cdot g(x) \) is \( 2 + n \).

2. **Degree comparison:**
   Since \( f(g(x)) \) must have the same degree as \( x^2 \cdot g(x) \), we have:
   \[
   \deg(f(g(x))) = m \cdot n = 2 + n
   \]
   This gives us the equation:
   \[
   mn = 2 + n
   \]

3. **Solve for \( m \):**
   Rearrange the equation to solve for \( m \):
   \[
   mn - n = 2 \implies n(m - 1) = 2 \implies m - 1 = \frac{2}{n} \implies m = 1 + \frac{2}{n}
   \]

4. **Consider possible values of \( n \):**
   Since \( m \) must be an integer, \( \frac{2}{n} \) must also be an integer. The possible values of \( n \) that satisfy this condition are \( n = 1 \) and \( n = 2 \).

5. **Case \( n = 1 \):**
   \[
   m = 1 + \frac{2}{1} = 3
   \]
   So, \( \deg(f) = 3 \) and \( \deg(g) = 1 \). Let \( g(x) = ax + b \). Then:
   \[
   x^2 \cdot (ax + b) = f(ax + b)
   \]
   Since \( \deg(f) = 3 \), let \( f(x) = px^3 + qx^2 + rx + s \). Substituting \( g(x) \) into \( f \):
   \[
   x^2 \cdot (ax + b) = p(ax + b)^3 + q(ax + b)^2 + r(ax + b) + s
   \]
   Equate the coefficients of \( x \) on both sides. This will give a system of equations that must be satisfied by the coefficients \( p, q, r, s, a, \) and \( b \).

6. **Case \( n = 2 \):**
   \[
   m = 1 + \frac{2}{2} = 2
   \]
   So, \( \deg(f) = 2 \) and \( \deg(g) = 2 \). Let \( g(x) = ax^2 + bx + c \). Then:
   \[
   x^2 \cdot (ax^2 + bx + c) = f(ax^2 + bx + c)
   \]
   Since \( \deg(f) = 2 \), let \( f(x) = px^2 + qx + r \). Substituting \( g(x) \) into \( f \):
   \[
   x^2 \cdot (ax^2 + bx + c) = p(ax^2 + bx + c)^2 + q(ax^2 + bx + c) + r
   \]
   Equate the coefficients of \( x \) on both sides. This will give a system of equations that must be satisfied by the coefficients \( p, q, r, a, b, \) and \( c \).

7. **Solve the system of equations:**
   For both cases, solving the system of equations will yield the specific forms of \( f \) and \( g \).

8. **Verification:**
   Verify the solutions by substituting back into the original equation to ensure they satisfy \( x^2 \cdot g(x) = f(g(x)) \).

The final answer is the pairs of polynomials \( \boxed{ (f, g) } \) that satisfy the derived systems of equations for each case.