To find the solutions to the system of equations:
[tex]\[
\begin{array}{l}
y = x^2 + 5x + 6 \\
y = 3x + 6
\end{array}
\][/tex]
we need to solve these equations simultaneously.
### Step-by-Step Solution:
1. Set the equations equal to each other:
[tex]\[
x^2 + 5x + 6 = 3x + 6
\][/tex]
2. Simplify the equation:
Subtract [tex]\(3x + 6\)[/tex] from both sides:
[tex]\[
x^2 + 5x + 6 - (3x + 6) = 0 \implies x^2 + 2x = 0
\][/tex]
3. Factor the quadratic equation:
[tex]\[
x(x + 2) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
The solutions to the factored equation are:
[tex]\[
x = 0 \quad \text{or} \quad x = -2
\][/tex]
5. Determine the corresponding [tex]\(y\)[/tex]-values for each [tex]\(x\)[/tex]:
- When [tex]\(x = 0\)[/tex]:
[tex]\[
y = 3x + 6 = 3(0) + 6 = 6
\][/tex]
Thus, one solution is [tex]\((0, 6)\)[/tex].
- When [tex]\(x = -2\)[/tex]:
[tex]\[
y = 3x + 6 = 3(-2) + 6 = -6 + 6 = 0
\][/tex]
Thus, the other solution is [tex]\((-2, 0)\)[/tex].
### Solutions:
The solutions to the system of equations are [tex]\((0, 6)\)[/tex] and [tex]\((-2, 0)\)[/tex]. Therefore, the correct answer is:
[tex]\[
(0,6) \text{ and } (-2,0)
\][/tex]
