To evaluate the expression \( 3x^2 + 4xy + 1 \) when \( x = 1 \) and \( y = -1 \):

1. Start with the given expression:  
   \[
   3x^2 + 4xy + 1
   \]

2. Substitute \( x = 1 \) and \( y = -1 \):  
   \[
   3(1)^2 + 4(1)(-1) + 1
   \]

3. Simplify each term:  
   - \( 3(1)^2 = 3 \)
   - \( 4(1)(-1) = -4 \)
   - The constant term is \( 1 \)

4. Now combine all the terms:  
   \[
   3 - 4 + 1 = 0
   \]

Thus, the value of the expression when \( x = 1 \) and \( y = -1 \) is \( \boxed{0} \).

Do you want further details or have any questions?

Here are some related questions:
1. What would the result be if \( x = 2 \) and \( y = 3 \)?
2. How does the expression change when \( x \) and \( y \) are both negative?
3. Can you factor the expression \( 3x^2 + 4xy + 1 \)?
4. What is the behavior of this expression when \( x \) and \( y \) approach infinity?
5. How would you solve for \( y \) if \( 3x^2 + 4xy + 1 = 0 \)?

**Tip:** Always substitute the values carefully and simplify step by step to avoid errors.

Learn how to evaluate the polynomial expression 3x² + 4xy + 1 by substituting x = 1 and y = -1. This problem is ideal for students in Grades 7-9, covering essential algebraic concepts.

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