To expand the expression $(b+6)(3b^2 - 8b - 14)$, we apply the distributive property:

$(b+6)(3b^2 - 8b - 14) = b(3b^2 - 8b - 14) + 6(3b^2 - 8b - 14).$

Step 1: Expand each term

1. $b(3b^2 - 8b - 14)$: $b \cdot 3b^2 = 3b^3, \quad b \cdot (-8b) = -8b^2, \quad b \cdot (-14) = -14b.$ Result: $3b^3 - 8b^2 - 14b$.

2. $6(3b^2 - 8b - 14)$: $6 \cdot 3b^2 = 18b^2, \quad 6 \cdot (-8b) = -48b, \quad 6 \cdot (-14) = -84.$ Result: $18b^2 - 48b - 84$.

Step 2: Combine like terms

Now add the results: $3b^3 - 8b^2 - 14b + 18b^2 - 48b - 84.$ Combine the $b^2$ terms and $b$ terms: $3b^3 + (-8b^2 + 18b^2) + (-14b - 48b) - 84 = 3b^3 + 10b^2 - 62b - 84.$

Final Answer:

$\boxed{3b^3 + 10b^2 - 62b - 84}$

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Relative Questions:

1. How does the distributive property work in polynomial expansions?
2. What are the steps to combine like terms in a polynomial?
3. Can you explain how to factorize the expanded polynomial?
4. What are real-world applications of polynomial multiplication?
5. How would the result change if the expression had different coefficients?

Tip:

When expanding polynomials, always distribute each term in the first expression to every term in the second expression systematically to avoid errors.