1. **State the problem:** We are given the quadratic equation $$4x^2 + 6xy + 2y^2 - 4x - 2y + 43 = 0$$ and we want to analyze or simplify it. 2. **Identify the type of equation:** This is a second-degree equation in two variables $x$ and $y$, representing a conic section. 3. **Rewrite the equation:** Group terms to complete the square if possible. 4. **Matrix form:** The quadratic form can be written as $$\begin{bmatrix}x & y\end{bmatrix} \begin{bmatrix}4 & 3 \\ 3 & 2\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} - 4x - 2y + 43 = 0$$ 5. **Complete the square:** To simplify, we can try to diagonalize the quadratic form or use substitution, but since the problem does not specify, we can leave it as is or analyze further if needed. 6. **Interpretation:** The equation represents a conic section (ellipse, hyperbola, or parabola) depending on the discriminant $B^2 - 4AC$ where $A=4$, $B=6$, $C=2$. Calculate discriminant: $$6^2 - 4 \times 4 \times 2 = 36 - 32 = 4 > 0$$ Since the discriminant is positive, the conic is a hyperbola. **Final answer:** The equation represents a hyperbola.