To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), we first rearrange the equation to bring all terms to one side:
\[ 10x^2 + 5x - 5x + 8 = 0 \]
This simplifies the equation to:
\[ 10x^2 + 8 = 0 \]
Next, we can isolate \( x^2 \) by moving \( 8 \) to the other side:
\[ 10x^2 = -8 \]
Now, we divide both sides by \( 10 \):
\[ x^2 = -\frac{8}{10} \] \[ x^2 = -\frac{4}{5} \]
Since we have a negative value under the square root, we will take the square root, leading us to complex solutions.
We can express \( x \) as follows:
\[ x = \pm \sqrt{-\frac{4}{5}} \]
This can be rewritten using the imaginary unit \( i \):
\[ x = \pm \sqrt{\frac{4}{5}} \cdot i \] \[ x = \pm \frac{2}{\sqrt{5}} \cdot i \] \[ x = \pm \frac{2\sqrt{5}}{5} i \]
Thus, the solutions of the equation \( 10x^2 + 8 = 0 \) expressed in the form \( a \pm bi \) are:
\[ 0 \pm \frac{2\sqrt{5}}{5} i \]
Finally, the solutions are:
\[ x = 0 + \frac{2\sqrt{5}}{5} i \quad \text{and} \quad x = 0 - \frac{2\sqrt{5}}{5} i \]
In the final answer, we can express it as:
\[ \boxed{0 \pm \frac{2\sqrt{5}}{5} i} \]