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The task involves multiplying the binomial \( (x-5) \) by the trinomial \( (x^2 - 2x + 6) \). We'll distribute \( (x-5) \) across each term in the trinomial.

\[
(x-5)(x^2 - 2x + 6) = x(x^2 - 2x + 6) - 5(x^2 - 2x + 6)
\]

Now, let's expand both products:

\[
x(x^2 - 2x + 6) = x^3 - 2x^2 + 6x
\]
\[
-5(x^2 - 2x + 6) = -5x^2 + 10x - 30
\]

Next, we'll combine like terms:

\[
x^3 - 2x^2 + 6x - 5x^2 + 10x - 30
\]
\[
= x^3 - (2x^2 + 5x^2) + (6x + 10x) - 30
\]
\[
= x^3 - 7x^2 + 16x - 30
\]

So, the product is \( x^3 - 7x^2 + 16x - 30 \).

Do you need more details on this calculation or do you have another question about the process?

### Further Explorations
1. What happens if you reverse the order of multiplication, does the answer change?
2. Can you factorize the resultant expression back to its original form?
3. How would adding a term to the binomial, making it a trinomial, change the calculation?
4. What is the general method for multiplying polynomials of any degree?
5. Could you apply the FOIL (First, Outer, Inner, Last) method here, or why might another method be more suitable?

**Tip:** Always line up like terms vertically when combining them after multiplication to avoid mistakes and ensure clarity in your work.

Learn how to multiply the polynomial expression (x - 5)(x^2 - 2x + 6) using the distributive property. This detailed step-by-step solution covers key algebraic concepts such as the distributive law and combining like terms. Suitable for students in Grades 8-10.

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