Suppose that $n^2-2m^2=m(n+3)-3$. Find all integers $m$ such that all corresponding solutions for $n$ will [i]not[/i] be real.
1. Start by rearranging the given equation \( n^2 - 2m^2 = m(n + 3) - 3 \) into a standard quadratic form in terms of \( n \):
   \[
   n^2 - mn + (3 - 3m - 2m^2) = 0
   \]

2. Identify the discriminant of the quadratic equation \( n^2 - mn + (3 - 3m - 2m^2) = 0 \). The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Here, \( a = 1 \), \( b = -m \), and \( c = 3 - 3m - 2m^2 \). Therefore, the discriminant is:
   \[
   \Delta = (-m)^2 - 4 \cdot 1 \cdot (3 - 3m - 2m^2)
   \]
   Simplify the expression:
   \[
   \Delta = m^2 - 4(3 - 3m - 2m^2)
   \]
   \[
   \Delta = m^2 - 4 \cdot 3 + 4 \cdot 3m + 4 \cdot 2m^2
   \]
   \[
   \Delta = m^2 - 12 + 12m + 8m^2
   \]
   \[
   \Delta = 9m^2 + 12m - 12
   \]

3. For \( n \) to be non-real, the discriminant must be less than 0:
   \[
   9m^2 + 12m - 12 < 0
   \]

4. Solve the inequality \( 9m^2 + 12m - 12 < 0 \) by finding the roots of the corresponding quadratic equation \( 9m^2 + 12m - 12 = 0 \):
   \[
   m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 9 \), \( b = 12 \), and \( c = -12 \):
   \[
   m = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 9 \cdot (-12)}}{2 \cdot 9}
   \]
   \[
   m = \frac{-12 \pm \sqrt{144 + 432}}{18}
   \]
   \[
   m = \frac{-12 \pm \sqrt{576}}{18}
   \]
   \[
   m = \frac{-12 \pm 24}{18}
   \]
   \[
   m = \frac{12}{18} \quad \text{or} \quad m = \frac{-36}{18}
   \]
   \[
   m = \frac{2}{3} \quad \text{or} \quad m = -2
   \]

5. The roots of the quadratic equation are \( m = \frac{2}{3} \) and \( m = -2 \). The quadratic \( 9m^2 + 12m - 12 \) is a parabola opening upwards (since the coefficient of \( m^2 \) is positive). Therefore, the inequality \( 9m^2 + 12m - 12 < 0 \) holds between the roots:
   \[
   -2 < m < \frac{2}{3}
   \]

6. The integers within this interval are \( m = -2, -1, 0 \).

Conclusion:
The integers \( m \) such that all corresponding solutions for \( n \) will not be real are:
\[
\boxed{\{-2, -1, 0\}}
\]