The given expression is:

\[
12 \sqrt{8x} + 2x \sqrt{4x}
\]

### Step 1: Simplify the square roots.
We can express \( \sqrt{8x} \) and \( \sqrt{4x} \) as:

\[
\sqrt{8x} = \sqrt{4 \times 2x} = \sqrt{4} \times \sqrt{2x} = 2\sqrt{2x}
\]
\[
\sqrt{4x} = \sqrt{4} \times \sqrt{x} = 2\sqrt{x}
\]

### Step 2: Substitute the simplified square roots back into the expression.

\[
12 \sqrt{8x} = 12 \times 2 \sqrt{2x} = 24 \sqrt{2x}
\]
\[
2x \sqrt{4x} = 2x \times 2 \sqrt{x} = 4x \sqrt{x}
\]

### Step 3: Combine the terms.
Now substitute both simplified terms into the original expression:

\[
24 \sqrt{2x} + 4x \sqrt{x}
\]

Thus, the simplified expression is:

\[
24 \sqrt{2x} + 4x \sqrt{x}
\]

Let me know if you'd like more details, or if you have any questions!

Here are some related problems for practice:
1. Simplify \( 10 \sqrt{18x} + 3x \sqrt{9x} \).
2. Solve \( 15 \sqrt{6x} - 5x \sqrt{3x} \).
3. Simplify \( 7 \sqrt{50x} + 5x \sqrt{25x} \).
4. Expand \( 8x \sqrt{16x} + 2x \sqrt{64x} \).
5. Factor \( 4 \sqrt{32x} + 6x \sqrt{2x} \).

**Tip**: Always simplify square roots by breaking them down into perfect squares before simplifying the overall expression.

Learn how to simplify the expression 12√8x + 2x√4x with a step-by-step guide. This problem covers key algebraic concepts including simplifying square roots and combining like terms. Ideal for students in Grades 9-11.

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