To solve the equation \( x + \sqrt{2} \cdot x^2 - 4 = 0 \), we can treat it as a standard quadratic equation of the form \( ax^2 + bx + c = 0 \).

Given:
\[ \sqrt{2} \cdot x^2 + x - 4 = 0 \]

Here, \( a = \sqrt{2} \), \( b = 1 \), and \( c = -4 \).

We can solve this using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

First, calculate the discriminant (\( \Delta \)):
\[ \Delta = b^2 - 4ac \]
\[ \Delta = 1^2 - 4(\sqrt{2})(-4) \]
\[ \Delta = 1 + 16\sqrt{2} \]

Now, apply the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{1 + 16\sqrt{2}}}{2\sqrt{2}} \]

So the solutions are:
\[ x = \frac{-1 + \sqrt{1 + 16\sqrt{2}}}{2\sqrt{2}} \]
\[ x = \frac{-1 - \sqrt{1 + 16\sqrt{2}}}{2\sqrt{2}} \]

These are the two solutions to the quadratic equation \( x + \sqrt{2} \cdot x^2 - 4 = 0 \).