https://file.aiquickdraw.com/mathbot/file/1724938871295ucv8g818.jpg

The system of equations given in the image is:

\[
\begin{cases}
9x^2 - 4y^2 + 54x + 45 = 0 \quad \text{(Equation 1)} \\
(x + 3)^2 = 4y + 4 \quad \text{(Equation 2)}
\end{cases}
\]

### Step 1: Simplify Equation 2

First, let's simplify Equation 2:
\[
(x + 3)^2 = 4y + 4
\]
Expanding the left-hand side:
\[
x^2 + 6x + 9 = 4y + 4
\]
Now, subtract 4 from both sides:
\[
x^2 + 6x + 5 = 4y \quad \text{(Equation 3)}
\]

### Step 2: Substitute Equation 3 into Equation 1

We can express \(4y\) from Equation 3 as:
\[
4y = x^2 + 6x + 5
\]
Substitute this into Equation 1:
\[
9x^2 - (x^2 + 6x + 5) + 54x + 45 = 0
\]
Simplify this equation:
\[
9x^2 - x^2 - 6x - 5 + 54x + 45 = 0
\]
\[
8x^2 + 48x + 40 = 0
\]
Now, divide the entire equation by 8 to simplify:
\[
x^2 + 6x + 5 = 0
\]

### Step 3: Solve the Quadratic Equation for \(x\)

The equation \(x^2 + 6x + 5 = 0\) is a quadratic equation. We can solve it using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 1\), \(b = 6\), and \(c = 5\):
\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(5)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 - 20}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{16}}{2}
\]
\[
x = \frac{-6 \pm 4}{2}
\]
Thus, the solutions for \(x\) are:
\[
x = \frac{-6 + 4}{2} = -1 \quad \text{and} \quad x = \frac{-6 - 4}{2} = -5
\]

### Step 4: Find Corresponding \(y\) Values

Now, substitute these \(x\) values back into Equation 3 to find the corresponding \(y\) values.

For \(x = -1\):
\[
4y = (-1)^2 + 6(-1) + 5 = 1 - 6 + 5 = 0
\]
\[
y = 0
\]

For \(x = -5\):
\[
4y = (-5)^2 + 6(-5) + 5 = 25 - 30 + 5 = 0
\]
\[
y = 0
\]

### Final Answer

The solutions to the system of equations are:
\[
(x, y) = (-1, 0) \quad \text{and} \quad (x, y) = (-5, 0)
\]

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you verify the solution of a system of nonlinear equations?
2. What methods can be used to solve a system of quadratic equations?
3. How does the quadratic formula help in solving quadratic equations?
4. What are the conditions for a system of equations to have no solution, one solution, or infinitely many solutions?
5. How can graphical methods be used to solve a system of equations?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure they satisfy both equations.

Learn how to solve a system of equations using elimination and substitution methods. This detailed guide includes step-by-step instructions and examples, focusing on quadratic equations and their solutions. Suitable for high school students studying algebra.

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