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For the question \( 7(b) \):

We are given the identity:

\[
(x - 7)(x + 2) \equiv x^2 + Ax - B
\]

### Step 1: Expand the left-hand side (LHS)
First, expand \( (x - 7)(x + 2) \):

\[
(x - 7)(x + 2) = x^2 + 2x - 7x - 14 = x^2 - 5x - 14
\]

So, the LHS is:

\[
LHS = x^2 - 5x - 14
\]

### Step 2: Compare with the right-hand side (RHS)
We are given the RHS as:

\[
RHS = x^2 + Ax - B
\]

By comparing the LHS and RHS, we can equate coefficients:

- For \( x^2 \), the coefficients are equal on both sides.
- For \( x \), we have \( -5 \) on the LHS, so \( A = -5 \).
- For the constant term, we have \( -14 \) on the LHS, so \( B = 14 \).

### Final Answer:
\[
A = -5, \quad B = 14
\]

Would you like more details or further explanation?

Here are 5 related questions to explore:
1. How do you expand the product of two binomials?
2. Can you explain how to compare coefficients in an identity?
3. What happens if \( A \) and \( B \) were fractions instead of integers?
4. How can you apply this method to identities involving cubic terms?
5. How would this problem change if additional terms, like \( Cx \), were present?

**Tip:** Always verify your expanded terms by multiplying them out again after simplifying, ensuring accuracy.

Learn how to solve the polynomial identity (x - 7)(x + 2) ≡ x^2 + Ax - B by expanding and comparing coefficients. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 8-10.

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