Solve by using midpoint method \( (\sqrt{2}, 3 \sqrt{5})(\sqrt{2}-2 \sqrt{5}) \)

To solve the expression \( (\sqrt{2}, 3 \sqrt{5})(\sqrt{2}-2 \sqrt{5}) \) using the midpoint method, we first need to clarify what the expression means. It seems to involve two components: a point or vector \( (\sqrt{2}, 3 \sqrt{5}) \) and a scalar \( (\sqrt{2} - 2 \sqrt{5}) \). Assuming that the expression represents a scalar multiplication of the vector by the scalar, we can proceed as follows: 1. **Identify the vector and scalar**: - Vector: \( \mathbf{v} = (\sqrt{2}, 3 \sqrt{5}) \) - Scalar: \( k = \sqrt{2} - 2 \sqrt{5} \) 2. **Perform the scalar multiplication**: The scalar multiplication of a vector \( \mathbf{v} = (x, y) \) by a scalar \( k \) is given by: \[ k \cdot \mathbf{v} = (k \cdot x, k \cdot y) \] Applying this to our vector: \[ k \cdot \mathbf{v} = (\sqrt{2} - 2 \sqrt{5}) \cdot (\sqrt{2}, 3 \sqrt{5}) \] This results in: \[ \left( (\sqrt{2} - 2 \sqrt{5}) \cdot \sqrt{2}, (\sqrt{2} - 2 \sqrt{5}) \cdot 3 \sqrt{5} \right) \] 3. **Calculate each component**: - For the first component: \[ (\sqrt{2} - 2 \sqrt{5}) \cdot \sqrt{2} = \sqrt{2} \cdot \sqrt{2} - 2 \sqrt{5} \cdot \sqrt{2} = 2 - 2\sqrt{10} \] - For the second component: \[ (\sqrt{2} - 2 \sqrt{5}) \cdot 3 \sqrt{5} = 3 \sqrt{5} \cdot \sqrt{2} - 6 \sqrt{5} \cdot \sqrt{5} = 3\sqrt{10} - 30 \] 4. **Combine the results**: Thus, the result of the scalar multiplication is: \[ \left( 2 - 2\sqrt{10}, 3\sqrt{10} - 30 \right) \] So, the final answer is: \[ \boxed{(2 - 2\sqrt{10}, 3\sqrt{10} - 30)} \]