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We are given the following equation:

\[
-(x + 1)(x - 4) \equiv A x^2 + 3x + B
\]

Our task is to expand the left-hand side and then compare it with the right-hand side to solve for \( A \) and \( B \).

### Step 1: Expand \( -(x + 1)(x - 4) \)

First, let's expand the product inside the parentheses:

\[
(x + 1)(x - 4) = x(x - 4) + 1(x - 4) = x^2 - 4x + x - 4 = x^2 - 3x - 4
\]

Now apply the negative sign:

\[
-(x^2 - 3x - 4) = -x^2 + 3x + 4
\]

### Step 2: Compare the expanded form with the right-hand side

Now, compare:

\[
-x^2 + 3x + 4 \equiv A x^2 + 3x + B
\]

From the comparison, we can see:
- The coefficient of \( x^2 \) is \( -1 \), so \( A = -1 \).
- The constant term is \( 4 \), so \( B = 4 \).

Thus, the values of \( A \) and \( B \) are:

\[
A = -1, \quad B = 4
\]

Let me know if you would like further details!

### Relative Questions:
1. How would the process change if the equation involved higher-degree polynomials?
2. What are other methods to expand polynomials like this one?
3. How do you solve for coefficients when the right-hand side involves more terms with \( x^2 \), \( x \), and constants?
4. How can this concept be applied in systems of equations with polynomials?
5. Why is it important to compare coefficients in polynomial identities?

### Tip:
When expanding polynomials, always apply distributive properties step by step, and don't forget to handle negative signs carefully when present.

Learn how to expand the polynomial expression -(x + 1)(x - 4) and compare coefficients to solve for A and B in the equation A x^2 + 3x + B. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 8-10.

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