To solve the equation
\[ 3\sqrt{x^2 - 10} + 7 = 4, \]
first, isolate the square root term. We do this by subtracting 7 from both sides:
\[ 3\sqrt{x^2 - 10} = 4 - 7, \] \[ 3\sqrt{x^2 - 10} = -3. \]
Next, divide both sides by 3:
\[ \sqrt{x^2 - 10} = -1. \]
Since the square root of a real number cannot be negative, there are no real solutions to this equation.
Thus, the result is:
\[ \text{No real solutions.} \]