https://file.aiquickdraw.com/mathbot/file/1726103255148ccmi6nux.jpg

The equation given in the image is:

\[
(y + 7)^2 = 2y^2 + 18y + 52
\]

Let's solve this step by step.

1. **Expand the left-hand side**:

\[
(y + 7)^2 = y^2 + 14y + 49
\]

Now, the equation becomes:

\[
y^2 + 14y + 49 = 2y^2 + 18y + 52
\]

2. **Move all terms to one side of the equation**:

\[
y^2 + 14y + 49 - 2y^2 - 18y - 52 = 0
\]

Simplify:

\[
-y^2 - 4y - 3 = 0
\]

Or multiply the equation by \(-1\) to make the leading coefficient positive:

\[
y^2 + 4y + 3 = 0
\]

3. **Factor the quadratic equation**:

\[
y^2 + 4y + 3 = (y + 1)(y + 3) = 0
\]

4. **Solve for \(y\)**:

\[
y + 1 = 0 \quad \text{or} \quad y + 3 = 0
\]

Thus, the solutions are:

\[
y = -1 \quad \text{and} \quad y = -3
\]

So, the two solutions are:

\[
y = -1, -3
\]

Would you like a detailed explanation on any specific step? 

Here are 5 questions that can expand this topic:
1. How do you know when to factor a quadratic equation?
2. What are alternative methods to solve a quadratic equation (besides factoring)?
3. How does completing the square work for solving quadratics?
4. Can quadratic equations always be factored into real solutions?
5. What are some applications of quadratic equations in real-world problems?

**Tip:** When solving quadratic equations, always check if the quadratic can be factored easily before using more advanced methods like the quadratic formula.

Learn how to solve the quadratic equation (y + 7)^2 = 2y^2 + 18y + 52 step-by-step. This problem demonstrates factoring and solving quadratics. Suitable for students in Grades 8-10.

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