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We are given the following equation:

\[
(x - 5)(2x + 1)(x + 3) \equiv ax^3 + bx^2 - 32x - 15
\]

### Steps to solve:

1. **Expand the left-hand side**:

   \[
   (x - 5)(2x + 1)(x + 3)
   \]

   Let's first expand the product of the first two binomials:

   \[
   (x - 5)(2x + 1) = 2x^2 - 9x - 5
   \]

2. **Multiply the result by the third binomial** \((x + 3)\):

   \[
   (2x^2 - 9x - 5)(x + 3) = 2x^3 + 6x^2 - 9x^2 - 27x - 5x - 15
   \]

   Simplifying:

   \[
   = 2x^3 - 3x^2 - 32x - 15
   \]

3. **Compare this to the given form** \(ax^3 + bx^2 - 32x - 15\):

   \[
   2x^3 - 3x^2 - 32x - 15
   \]

   By comparing coefficients, we get:
   - Coefficient of \(x^3\): \(a = 2\)
   - Coefficient of \(x^2\): \(b = -3\)

Thus, the values are:

\[
a = 2, \quad b = -3
\]

---

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How would the values of \(a\) and \(b\) change if one of the factors was different?
2. How do the coefficients of each term affect the final equation?
3. What happens if one of the constants in the binomials is zero?
4. Can you factor the resulting cubic polynomial back into binomials?
5. How can you verify that the expanded form is correct?

**Tip**: Always recheck by expanding again to ensure the values you derived are consistent with the original equation.

Given that (x - 5)(2x + 1)(x + 3) ≡ ax^3 + bx^2 - 32x - 15 where a and b are constants, find the value of a and the value of b.

Learn how to expand the product of three binomials to find the constants a and b in the cubic equation ax^3 + bx^2 - 32x - 15. This solution involves key algebraic concepts like polynomial expansion and comparison of coefficients. Suitable for students in Grades 10-12.

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