To factor the quadratic expression \(4p^2 - 5p - 6\), follow these steps:

### Step 1: Multiply the leading coefficient by the constant term.
Multiply the coefficient of \(p^2\) (which is 4) by the constant term (which is -6):
\[
4 \times -6 = -24
\]

### Step 2: Find two numbers that multiply to -24 and add to the middle term (-5).
We need two numbers that multiply to -24 and add to -5. The numbers are -8 and 3, because:
\[
-8 \times 3 = -24 \quad \text{and} \quad -8 + 3 = -5
\]

### Step 3: Rewrite the middle term using these two numbers.
Rewrite the expression as:
\[
4p^2 - 8p + 3p - 6
\]

### Step 4: Factor by grouping.
Group the terms in pairs:
\[
(4p^2 - 8p) + (3p - 6)
\]
Factor each group:
\[
4p(p - 2) + 3(p - 2)
\]

### Step 5: Factor out the common binomial factor.
Since both groups contain \((p - 2)\), factor it out:
\[
(4p + 3)(p - 2)
\]

### Final factored form:
\[
4p^2 - 5p - 6 = (4p + 3)(p - 2)
\]

Would you like any additional explanations or have any questions?

Here are 5 related questions you might find useful:
1. How do you factor trinomials with higher degrees?
2. Can you explain the process of factoring by grouping in more detail?
3. How do you factor perfect square trinomials?
4. What is the quadratic formula, and how does it apply to this type of equation?
5. How do you solve quadratic equations by factoring?

**Tip**: Always check your factorization by expanding the factors to see if you return to the original expression.

Learn how to factor the quadratic expression 4p² - 5p - 6 step-by-step using the factoring by grouping method. This guide explains how to find two numbers that multiply to -24 and add to -5, and how to group and factor the terms. Ideal for students in Grades 9-10 studying algebra.

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