Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$  $x, y, z$  $\in \mathbb{R}$
$x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.
1. **Initial Setup:**
   We need to find the greatest value of \( M \in \mathbb{R} \) such that the inequality 
   \[
   x^4 + y^4 + z^4 + xyz(x + y + z) \geq M(xy + yz + zx)^2
   \]
   holds for all \( x, y, z \in \mathbb{R} \).

2. **Testing for \( x = y = z \):**
   Let's start by setting \( x = y = z \). In this case, the inequality becomes:
   \[
   3x^4 + 3x^4 \geq M(3x^2)^2
   \]
   Simplifying, we get:
   \[
   6x^4 \geq 9Mx^4
   \]
   Dividing both sides by \( x^4 \) (assuming \( x \neq 0 \)):
   \[
   6 \geq 9M
   \]
   Therefore:
   \[
   M \leq \frac{2}{3}
   \]

3. **Proving the Inequality for \( M = \frac{2}{3} \):**
   Now, we need to prove that the inequality holds when \( M = \frac{2}{3} \). The inequality becomes:
   \[
   x^4 + y^4 + z^4 + xyz(x + y + z) \geq \frac{2}{3}(xy + yz + zx)^2
   \]
   Multiplying both sides by 3, we get:
   \[
   3(x^4 + y^4 + z^4) + 3xyz(x + y + z) \geq 2(xy + yz + zx)^2
   \]

4. **Breaking Down the Inequality:**
   We need to show:
   \[
   3(x^4 + y^4 + z^4) \geq 2(x^2y^2 + y^2z^2 + z^2x^2)
   \]
   and
   \[
   3xyz(x + y + z) \geq 0
   \]

5. **First Part:**
   Consider the expression \( \sum x^4 - \sum x^2y^2 \):
   \[
   \sum x^4 - \sum x^2y^2 = \frac{1}{2} \sum (x^2 - y^2)^2 \geq 0
   \]
   This shows that:
   \[
   x^4 + y^4 + z^4 \geq x^2y^2 + y^2z^2 + z^2x^2
   \]
   Therefore:
   \[
   3(x^4 + y^4 + z^4) \geq 3(x^2y^2 + y^2z^2 + z^2x^2)
   \]

6. **Second Part:**
   Consider the expression \( \sum x^2y^2 - xyz \sum x \):
   \[
   \sum x^2y^2 - xyz \sum x = \frac{1}{2} \sum z^2(x - y)^2 \geq 0
   \]
   This shows that:
   \[
   x^2y^2 + y^2z^2 + z^2x^2 \geq xyz(x + y + z)
   \]

7. **Combining Both Parts:**
   Adding the two parts together, we get:
   \[
   3(x^4 + y^4 + z^4) + 3xyz(x + y + z) \geq 2(x^2y^2 + y^2z^2 + z^2x^2)
   \]
   Hence, the inequality holds for \( M = \frac{2}{3} \).

8. **Conclusion:**
   Therefore, the greatest value of \( M \) such that the inequality holds for all \( x, y, z \in \mathbb{R} \) is \( \frac{2}{3} \).

The final answer is \( \boxed{\frac{2}{3}} \).