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We are given the quadratic expression:

\[
20x^2 - 19x - 28
\]

To factor it completely, we'll use the "AC method" (also known as factoring by grouping). Here's the step-by-step process:

### Step 1: Multiply the coefficient of \(x^2\) and the constant term
The coefficient of \(x^2\) is \(20\) and the constant term is \(-28\). Multiply these two together:

\[
20 \times (-28) = -560
\]

### Step 2: Find two numbers that multiply to \(-560\) and add to the middle coefficient, \(-19\)
We need two numbers that multiply to \(-560\) and add to \(-19\). The numbers are \(16\) and \(-35\):

\[
16 \times (-35) = -560 \quad \text{and} \quad 16 + (-35) = -19
\]

### Step 3: Rewrite the middle term using these two numbers
We rewrite the expression by splitting the middle term, \(-19x\), into \(16x\) and \(-35x\):

\[
20x^2 + 16x - 35x - 28
\]

### Step 4: Factor by grouping
Now, group the terms in pairs:

\[
(20x^2 + 16x) - (35x + 28)
\]

Factor out the greatest common factor (GCF) from each group:

\[
4x(5x + 4) - 7(5x + 4)
\]

### Step 5: Factor out the common binomial
Both groups contain the common factor \((5x + 4)\), so factor it out:

\[
(5x + 4)(4x - 7)
\]

### Final Answer:
The completely factored form of the expression is:

\[
(5x + 4)(4x - 7)
\]

Would you like more details on any step?

### Follow-up Questions:
1. What are the general steps to factor a quadratic expression?
2. Can all quadratics be factored using the AC method?
3. How can you verify that your factoring is correct?
4. What is the significance of factoring in solving quadratic equations?
5. What happens if no integers satisfy the factoring condition?

### Tip:
Always check your factoring by expanding the factors back to ensure they give the original expression.

Factor the quadratic expression: 20x^2 - 19x - 28.

Learn how to factor the quadratic expression 20x^2 - 19x - 28 using the AC method and grouping. Step-by-step solution with key algebraic concepts. Ideal for students in Grades 9-11.

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