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The equation provided is:

\[
5\sqrt{5}x^2 + 30x + 8\sqrt{5}
\]

To factor this by splitting the middle term, let's go through the steps:

### Step 1: Multiply the coefficient of \(x^2\)-term and the constant term.
- Coefficient of \(x^2\)-term = \(5\sqrt{5}\)
- Constant term = \(8\sqrt{5}\)

So, the product is:

\[
5\sqrt{5} \times 8\sqrt{5} = 40 \times 5 = 200
\]

### Step 2: Find two numbers that multiply to give 200 and add to give the middle term coefficient (which is 30).
- The numbers are 20 and 10 because:
  - \(20 \times 10 = 200\)
  - \(20 + 10 = 30\)

### Step 3: Split the middle term using these numbers.
\[
5\sqrt{5}x^2 + 20x + 10x + 8\sqrt{5}
\]

### Step 4: Factor by grouping.
Group the terms as follows:

\[
(5\sqrt{5}x^2 + 20x) + (10x + 8\sqrt{5})
\]

Now factor out the common factors:
\[
5x(\sqrt{5}x + 4) + 2(\sqrt{5}x + 4)
\]

### Step 5: Factor out the common binomial factor.
\[
(5x + 2)(\sqrt{5}x + 4)
\]

Thus, the factorized form is:

\[
(5x + 2)(\sqrt{5}x + 4)
\]

Would you like more details or have any questions? 

Here are 5 related questions to expand understanding:
1. How do you know when to use the method of splitting the middle term?
2. Can this method be used for all quadratic equations?
3. Why do we factor by grouping in the middle-term splitting method?
4. What other methods can be used to factor quadratic equations?
5. How do irrational coefficients, like \(\sqrt{5}\), affect the factorization process?

**Tip:** Always check if you can factor out a greatest common factor (GCF) before proceeding with splitting the middle term, as it simplifies the process.

Factorise: 5√5x² + 30x + 8√5 by splitting the middle term.

This problem teaches how to factorize the quadratic equation 5√5x² + 30x + 8√5 by splitting the middle term. Learn about important algebraic concepts such as factoring by grouping and using irrational coefficients. Suitable for students in Grades 10-12.

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