Start by expanding the right-hand side of the equation:
\[
y + 3 = -\frac{1}{3}(2x + 6)
\]
Distribute \(-\frac{1}{3}\) to both terms inside the parentheses:
\[
y + 3 = -\frac{1}{3} \cdot 2x - \frac{1}{3} \cdot 6
\]
Simplify the multiplication:
\[
y + 3 = -\frac{2}{3}x - 2
\]
Step 2: Move all terms to one side
To write the equation in standard form \(Ax + By + C = 0\), move all terms to one side:
\[
y + 3 + \frac{2}{3}x + 2 = 0
\]
Combine like terms:
\[
\frac{2}{3}x + y + 5 = 0
\]
Step 3: Eliminate fractions
Multiply every term by 3 to eliminate the fraction:
\[
3 \cdot \frac{2}{3}x + 3 \cdot y + 3 \cdot 5 = 0
\]
Simplify:
\[
2x + 3y + 15 = 0
\]
Final Answer
The equation in standard form is:
\[
\boxed{2x + 3y + 15 = 0}
\]