(-3sqrt (2)+sqrt (5))(sqrt (2)+3sqrt (5))

To solve the expression \((-3\sqrt{2} + \sqrt{5})(\sqrt{2} + 3\sqrt{5})\), we can use the distributive property (also known as the FOIL method for binomials):<br /><br />\[<br />(-3\sqrt{2} + \sqrt{5})(\sqrt{2} + 3\sqrt{5})<br />\]<br /><br />First, distribute \(-3\sqrt{2}\):<br /><br />\[<br />-3\sqrt{2} \cdot \sqrt{2} + (-3\sqrt{2}) \cdot 3\sqrt{5}<br />\]<br /><br />This simplifies to:<br /><br />\[<br />-3(\sqrt{2} \cdot \sqrt{2}) - 9\sqrt{2} \cdot \sqrt{5}<br />\]<br /><br />Since \(\sqrt{2} \cdot \sqrt{2} = 2\), we get:<br /><br />\[<br />-3 \cdot 2 - 9\sqrt{10}<br />\]<br /><br />This simplifies to:<br /><br />\[<br />-6 - 9\sqrt{10}<br />\]<br /><br />Next, distribute \(\sqrt{5}\):<br /><br />\[<br />\sqrt{5} \cdot \sqrt{2} + \sqrt{5} \cdot 3\sqrt{5}<br />\]<br /><br />This simplifies to:<br /><br />\[<br />\sqrt{10} + 3(\sqrt{5} \cdot \sqrt{5})<br />\]<br /><br />Since \(\sqrt{5} \cdot \sqrt{5} = 5\), we get:<br /><br />\[<br />\sqrt{10} + 3 \cdot 5<br />\]<br /><br />This simplifies to:<br /><br />\[<br />\sqrt{10} + 15<br />\]<br /><br />Now, combine all the terms:<br /><br />\[<br />-6 - 9\sqrt{10} + \sqrt{10} + 15<br />\]<br /><br />Combine the like terms:<br /><br />\[<br />-6 + 15 - 9\sqrt{10} + \sqrt{10}<br />\]<br /><br />This simplifies to:<br /><br />\[<br />9 - 8\sqrt{10}<br />\]<br /><br />So, the final answer is:<br /><br />\[<br />9 - 8\sqrt{10}<br />\]