To find the number of real solutions for the equation
\[ -15x^2 - 3 = 2(7x^2 - 1.5), \]
first, let's simplify the right side of the equation:
\[ 2(7x^2 - 1.5) = 14x^2 - 3. \]
Now we can rewrite the equation as:
\[ -15x^2 - 3 = 14x^2 - 3. \]
Next, we'll move all terms to one side of the equation by adding \(15x^2\) and 3 to both sides:
\[ 0 = 14x^2 - 3 + 15x^2 + 3. \]
This simplifies to:
\[ 0 = (14x^2 + 15x^2) + (-3 + 3), \] \[ 0 = 29x^2. \]
Now, we can solve for \(x^2\):
\[ 29x^2 = 0. \]
Dividing both sides by 29 gives:
\[ x^2 = 0. \]
This implies:
\[ x = 0. \]
Therefore, there is exactly one real solution to the equation, which is \(x = 0\).
In conclusion, the number of real solutions to the equation is
\[ \boxed{1}. \]