To solve the equation \( (1 - xy)y^2 + y^2 + 3xy^3 = 0 \), let's simplify and solve it step by step.

### Step 1: Simplify the Equation
First, factor out the common term \( y^2 \) from the equation:

\[
y^2 \left[(1 - xy) + 1 + 3xy\right] = 0
\]

### Step 2: Simplify the Expression Inside the Brackets
Combine like terms inside the brackets:

\[
(1 - xy + 1 + 3xy) = (2 + 2xy)
\]

So the equation becomes:

\[
y^2 (2 + 2xy) = 0
\]

### Step 3: Analyze the Factors
For the product to be zero, either of the factors must be zero:

1. \( y^2 = 0 \)
2. \( 2 + 2xy = 0 \)

#### Case 1: \( y^2 = 0 \)
If \( y^2 = 0 \), then \( y = 0 \).

#### Case 2: \( 2 + 2xy = 0 \)
If \( 2 + 2xy = 0 \), then:

\[
2xy = -2
\]
\[
xy = -1
\]

### Final Solutions
The solutions to the equation are:

1. \( y = 0 \)
2. \( xy = -1 \)

### Summary
- \( y = 0 \) is a solution.
- \( xy = -1 \) is another solution, which implies \( x = -\frac{1}{y} \) if \( y \neq 0 \).

Would you like any more details or have any questions?

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**Related Questions:**

1. What would the solution be if we had \( (1 - xy)y^2 + y^2 + 3xy^2 = 0 \) instead?
2. How does the solution change if the equation was \( (1 - xy)y^2 + y^3 + 3xy^3 = 0 \)?
3. What happens if the equation is multiplied by a non-zero constant?
4. How do the solutions change if \( x \) and \( y \) are restricted to integer values?
5. Can we interpret the solutions graphically, and what would the curves look like?

**Tip:** Always check for common factors in an equation, as factoring can significantly simplify the problem.

Learn how to solve the equation (1 - xy)y^2 + y^2 + 3xy^3 = 0 with detailed step-by-step solutions. This problem involves algebraic manipulation, factoring, and solving equations involving variables. Suitable for advanced high school students and beyond.

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