factor $$ {\displaylines4x^2-17x-15\\ } $$

The given quadratic expression is:

\[
4x^2 - 17x - 15
\]

To factor this quadratic expression, we'll use the method of **splitting the middle term**. Here's the process:

1. **Multiply the coefficient of $ x^2 $** (which is 4) by the constant term (which is $-15$):

   \[
   4 \times (-15) = -60
   \]

2. **Find two numbers that multiply to $-60$** and add to $-17$** (the coefficient of $ x $):

   The numbers are $-20$ and $3$, since:
   \[
   -20 \times 3 = -60 \quad \text{and} \quad -20 + 3 = -17
   \]

3. **Rewrite the middle term** ($-17x$) using these two numbers:

   \[
   4x^2 - 20x + 3x - 15
   \]

4. **Factor by grouping**:
   - Group the terms in pairs:
   \[
   (4x^2 - 20x) + (3x - 15)
   \]
   - Factor out the common factors from each group:
   \[
   4x(x - 5) + 3(x - 5)
   \]

5. **Factor out the common binomial** $ (x - 5) $:

   \[
   (4x + 3)(x - 5)
   \]

Thus, the factorization of $ 4x^2 - 17x - 15 $ is:

\[
(4x + 3)(x - 5)
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would you factor the quadratic expression $ 4x^2 + 17x + 15 $?
2. Can you factor $ 4x^2 - 20x + 25 $?
3. What is the discriminant of the quadratic equation $ 4x^2 - 17x - 15 = 0 $?
4. How do you solve $ 4x^2 - 17x - 15 = 0 $ using the quadratic formula?
5. Can the expression $ 4x^2 - 9x - 12 $ be factored using a similar method?

**Tip:** When factoring quadratic expressions, always look for common factors and use methods like splitting the middle term for easier computation.

Learn how to factor the quadratic expression 4x^2 - 17x - 15 using the splitting the middle term method. This detailed step-by-step solution covers key algebraic concepts and formulas for factoring quadratic equations. Suitable for high school students.

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