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Let's solve question 7(a) step by step:

We are given the equation:

\[
(x + 5)(x - 3) \equiv x^2 + Ax + B
\]

### Step 1: Expand the left-hand side (LHS)

We will first expand the expression \((x + 5)(x - 3)\):

\[
(x + 5)(x - 3) = x(x - 3) + 5(x - 3)
\]
\[
= x^2 - 3x + 5x - 15
\]
\[
= x^2 + 2x - 15
\]

So, the LHS becomes:

\[
x^2 + 2x - 15
\]

### Step 2: Compare with the right-hand side (RHS)

Now we compare this with the given right-hand side:

\[
x^2 + Ax + B
\]

By comparing the coefficients of \(x^2\), \(x\), and the constant term on both sides, we get:

- The coefficient of \(x^2\) is already \(1\) on both sides, so no comparison is needed for that.
- The coefficient of \(x\) is \(2\) on the left-hand side, so \(A = 2\).
- The constant term on the left-hand side is \(-15\), so \(B = -15\).

### Final Answer:
- \(A = 2\)
- \(B = -15\)

Let me know if you would like any further details or explanations!

---

Here are five related questions for further practice:

1. How do you expand the product of two binomials like \((x + 4)(x - 2)\)?
2. What is the general form of a quadratic equation after expanding \((x + m)(x + n)\)?
3. How would the values of \(A\) and \(B\) change if the equation was \((x + 6)(x - 4) \equiv x^2 + Ax + B\)?
4. Can you derive \(A\) and \(B\) for a different pair of constants in a similar equation?
5. How can you use factoring to reverse the expansion process of a quadratic expression?

**Tip**: Always compare coefficients carefully when solving equations involving identities like this one!

Step-by-step solution for expanding and solving the quadratic identity (x + 5)(x - 3) ≡ x² + Ax + B. This solution covers the binomial expansion process and coefficient comparison. Suitable for students in Grades 8-10.

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