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To solve the expression \((5 + 5\sqrt{x})^2\) as a trinomial, we need to apply the binomial square formula, which is:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Here, \(a = 5\) and \(b = 5\sqrt{x}\). Let's break it down step by step:

1. **Square the first term:**
   \[
   a^2 = 5^2 = 25
   \]

2. **Multiply the first and second terms by 2:**
   \[
   2ab = 2(5)(5\sqrt{x}) = 50\sqrt{x}
   \]

3. **Square the second term:**
   \[
   b^2 = (5\sqrt{x})^2 = 25x
   \]

Now, combine these results to express the trinomial:

\[
25 + 50\sqrt{x} + 25x
\]

Thus, the expanded trinomial is:

\[
25 + 50\sqrt{x} + 25x
\]

Let me know if you'd like any additional details or have questions. Here are five related problems to deepen your understanding:

1. Expand \((3 + 2\sqrt{y})^2\).
2. Express \((4 + \sqrt{z})^2\) as a trinomial.
3. Simplify \((7 + 3\sqrt{x})^2\) using the same method.
4. Expand \((2 + \sqrt{3x})^2\) as a trinomial.
5. Solve for \(x\) in \((6 + 4\sqrt{x})^2 = 100\).

**Tip:** When squaring binomials with square roots, always apply the binomial expansion formula carefully to avoid missing terms!

Distribute and express as a trinomial: (5 + 5√x)^2

Learn how to expand the binomial (5 + 5√x)^2 step by step using the binomial theorem. This problem covers algebraic concepts such as binomial expansion and handling square roots. Suitable for students in Grades 9-11.

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