Multiply and simplify. $$ \begin{array}{l}(\sqrt{x}+2 \sqrt{5})^{2}=\square \ (\sqrt{x}-\sqrt{5})(\sqrt{x}+\sqrt{5})=\square\end{array} $$

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UpStudy AI · Accepted Answer

Sure, let's multiply and simplify each expression step by step.

### 1. Expand $$(\sqrt{x} + 2\sqrt{5})^2$$

To expand the square of a binomial $$(a + b)^2$$, use the formula:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$

Applying this to $$(\sqrt{x} + 2\sqrt{5})^2$$:
$$
\begin{align*}
(\sqrt{x} + 2\sqrt{5})^2 &= (\sqrt{x})^2 + 2 \cdot \sqrt{x} \cdot 2\sqrt{5} + (2\sqrt{5})^2 \
&= x + 4\sqrt{5x} + 4 \cdot 5 \
&= x + 4\sqrt{5x} + 20 \
&= x + 20 + 4\sqrt{5x}
\end{align*}
$$

### 2. Multiply $$(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})$$

This is a difference of squares, which follows the formula:
$$
(a - b)(a + b) = a^2 - b^2
$$

Applying this to $$(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})$$:
$$
\begin{align*}
(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5}) &= (\sqrt{x})^2 - (\sqrt{5})^2 \
&= x - 5
\end{align*}
$$

### **Final Simplified Forms:**
$$
\begin{array}{l}
(\sqrt{x} + 2\sqrt{5})^2 = x + 20 + 4\sqrt{5x} \
(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5}) = x - 5
\end{array}
$$
$$ \begin{array}{l} (\sqrt{x} + 2\sqrt{5})^2 = x + 20 + 4\sqrt{5x} \ (\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5}) = x - 5 \end{array} $$

Multiply and simplify. \( \begin{array}{l}(\sqrt{x}+2 \sqrt{5})^{2}=\square \\ (\sqrt{x}-\sqrt{5})(\sqrt{x}+\sqrt{5})=\square\end{array} \)

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