The problem asks us to solve the equation $2y^2 + 10y = -9$ by completing the square.

1. Problem Description

The problem asks us to solve the equation $2y^2 + 10y = -9$ by completing the square.

2. Solution Steps

First, divide both sides of the equation by 2 to make the coefficient of the $y^2$ term equal to 1:

$y^2 + 5y = -\frac{9}{2}$

To complete the square, we need to add $(\frac{5}{2})^2 = \frac{25}{4}$ to both sides of the equation:

$y^2 + 5y + \frac{25}{4} = -\frac{9}{2} + \frac{25}{4}$

Rewrite the left side as a perfect square:

$(y + \frac{5}{2})^2 = -\frac{18}{4} + \frac{25}{4}$

$(y + \frac{5}{2})^2 = \frac{7}{4}$

Take the square root of both sides:

$y + \frac{5}{2} = \pm \sqrt{\frac{7}{4}}$

$y + \frac{5}{2} = \pm \frac{\sqrt{7}}{2}$

Isolate $y$:

$y = -\frac{5}{2} \pm \frac{\sqrt{7}}{2}$

Combine the terms:

$y = \frac{-5 \pm \sqrt{7}}{2}$

3. Final Answer

The solutions are $y = \frac{-5 + \sqrt{7}}{2}$ and $y = \frac{-5 - \sqrt{7}}{2}$. Therefore, the final answer is:

$y = \frac{-5 \pm \sqrt{7}}{2}$