\[ (3)(2) + (3)(\sqrt{5}) + (\sqrt{5})(2) + (\sqrt{5})(\sqrt{5}) = 6 + 3\sqrt{5} + 2\sqrt{5} + 5 = 11 + 5\sqrt{5} \]

["Mastering Algebraic Simplification: Solving (3)(2) + (3)√5 + (√5)(2) + (√5)(√5) Step by Step", "When tackling complex algebraic expressions, breaking them down step by step is essential for accuracy and clarity. One frequently encountered expression involves both rational and radical components:", "[\n(3)(2) + (3)\sqrt{5} + (\sqrt{5})(2) + (\sqrt{5})(\sqrt{5})\n]", "This equation combines multiplication of integers with irrational numbers like √5. Understanding how to simplify such expressions unlocks powerful problem-solving skills in algebra, calculus, and beyond.", "### Step 1: Identify Each Term", "Let’s analyze the expression term by term:", "- ((3)(2)): product of two rational numbers\n- ((3)\sqrt{5}): rational times irrational\n- ((\sqrt{5})(2)): irrational times rational\n- ((\sqrt{5})(\sqrt{5})): product of two square roots", "### Step 2: Apply Basic Multiplication Rules", "Recall that √5 × √5 = 5, a fundamental identity derived from the definition of square roots. Using this, simplify the last term:", "[\n(\sqrt{5})(\sqrt{5}) = 5\n]", "Now substitute that back into the expression:", "[\n(3)(2) + (3)\sqrt{5} + (\sqrt{5})(2) + 5\n]", "### Step 3: Multiply Remaining Rational and Irrational Products", "Next, compute the products:", "- ((3)(2) = 6)\n- ((\sqrt{5})(2) = 2\sqrt{5})", "So now the expression becomes:", "[\n6 + 3\sqrt{5} + 2\sqrt{5} + 5\n]", "### Step 4: Combine Like Terms", "Group rational and irrational parts:", "- Rational terms: (6 + 5 = 11)\n- Radical terms: (3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5})", "Thus, the fully simplified expression is:", "[\n11 + 5\sqrt{5}\n]", "### Why This Simplification Matters", "This process demonstrates how algebraic expressions can gracefully combine integers and radicals. Proper simplification ensures clarity, especially when substituting values or solving equations involving square roots. It’s also a foundational skill for higher-level math like quadratic equations, polynomial identities, and even calculus derivatives involving radicals.", "### Final Answer", "[\n\boxed{11 + 5\sqrt{5}}\n]", "Understanding such algebraic simplifications builds confidence and precision—key traits for success in chemistry, physics, engineering, and computer science where algebraic manipulation is constant. Keep practicing step-by-step decomposition, and algebraic expressions become intuitive and effortless."]