Simplify the following.
$\sqrt{5x}(4+\sqrt{x})$

• A. $4\sqrt{5x}+\sqrt{10x}$
• B. $4\(\sqrt{5x}\)+x\(\sqrt\)5$
• C. $10\(\sqrt{x}\)+x\(\sqrt\)5$
• D. $10\(\sqrt{x}\)+\(\sqrt{10x}\)$

Answer: (B)

Verified step by step guidance

1. Start with the expression \(\sqrt{5x}(4 + \sqrt{x})\). The goal is to simplify this by distributing the square root term over the sum inside the parentheses.
2. Distribute \(\sqrt{5x}\) to both terms inside the parentheses: multiply \(\sqrt{5x}\) by 4, and then multiply \(\sqrt{5x}\) by \(\sqrt{x}\). This gives \(4\sqrt{5x} + \sqrt{5x} \cdot \sqrt{x}\).
3. Recall the property of square roots: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Use this to combine \(\sqrt{5x} \cdot \sqrt{x}\) into \(\sqrt{5x \cdot x} = \sqrt{5x^2}\).
4. Simplify \(\sqrt{5x^2}\) by separating the perfect square: \(\sqrt{5x^2} = \sqrt{5} \cdot \sqrt{x^2}\). Since \(\sqrt{x^2} = x\), rewrite this as \(x \sqrt{5}\).
5. Rewrite the entire expression as \(4\sqrt{5x} + x\sqrt{5}\). This is the simplified form of the original expression.