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Let's solve the problem \( 8(a) \):

We are given the identity:

\[
2(x^2 - 1) \equiv Ax^2 + Bx + C
\]

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

\[
2(x^2 - 1) = 2x^2 - 2
\]

So, the LHS is:

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

### Step 2: Compare both sides of the equation

Now, we compare it with the right-hand side (RHS):

\[
Ax^2 + Bx + C
\]

Matching the terms from both sides:
- The coefficient of \(x^2\): \(A = 2\)
- The coefficient of \(x\): There is no \(x\) term on the left-hand side, so \(B = 0\)
- The constant term: \(C = -2\)

### Final Answer:

\[
A = 2, \quad B = 0, \quad C = -2
\]

Would you like to go over the steps in more detail or solve another part? Here are five related questions:

1. What happens if we change the quadratic term in the given equation?
2. How would the solution change if the constant on the left-hand side were positive?
3. Can you find similar identities for cubic polynomials?
4. How do we approach identities with more variables?
5. What is the general strategy to solve equations involving unknown constants?

**Tip:** Always compare the coefficients of like terms when solving identities.

Solve the identity 2(x^2 - 1) ≡ Ax^2 + Bx + C. Find the values of A, B, and C.

Learn how to solve the quadratic identity 2(x^2 - 1) ≡ Ax^2 + Bx + C. This problem demonstrates how to expand and compare coefficients to find unknown constants A, B, and C. Ideal for students in Grades 9-11.

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