To factor \( 4x^2 - 17x + 4 \) by splitting the middle term, we follow these steps:

### Step 1: Multiply the coefficient of \(x^2\) (which is 4) by the constant term (which is also 4).
\[
4 \times 4 = 16
\]

### Step 2: Find two numbers that multiply to 16 and add up to -17 (the middle term).
The two numbers are -16 and -1 because:
\[
-16 \times -1 = 16 \quad \text{and} \quad -16 + (-1) = -17
\]

### Step 3: Rewrite the middle term using -16 and -1.
\[
4x^2 - 16x - x + 4
\]

### Step 4: Factor by grouping.
Group the terms in pairs:
\[
(4x^2 - 16x) - (x - 4)
\]

Factor out the common factors in each group:
\[
4x(x - 4) - 1(x - 4)
\]

### Step 5: Factor out the common binomial factor.
\[
(4x - 1)(x - 4)
\]

Thus, the factorization of \(4x^2 - 17x + 4\) is:
\[
(4x - 1)(x - 4)
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you handle cases when there are no real factors for a quadratic?
2. What happens if the middle term cannot be split into integers?
3. Can you factor \(6x^2 - 5x - 4\) using the same method?
4. How would you solve \(4x^2 - 17x + 4 = 0\) after factoring it?
5. What is the significance of factoring quadratics in solving algebraic equations?

**Tip**: When choosing numbers to split the middle term, make sure they multiply to the product of the first and last coefficients, and add to the middle coefficient.

Learn how to factor the quadratic expression 4x^2 - 17x + 4 using the split-the-middle term method. This step-by-step solution covers key algebraic concepts such as factoring quadratics and finding product terms. Ideal for students in Grades 8-10.

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