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The expression shown is:

\[
x(3 - x)(7 - 3x) - (x^2 - 1)(3x + 1)
\]

Let's simplify this step-by-step.

### Step 1: Expand \( x(3 - x)(7 - 3x) \)
1. First, distribute \( x \) through \( (3 - x)(7 - 3x) \).
2. Calculate \( (3 - x)(7 - 3x) \) using the distributive property:
   \[
   (3 - x)(7 - 3x) = 3 \cdot 7 + 3 \cdot (-3x) - x \cdot 7 - x \cdot (-3x)
   \]
   \[
   = 21 - 9x - 7x + 3x^2
   \]
   \[
   = 3x^2 - 16x + 21
   \]
3. Now, multiply \( x \) through the result:
   \[
   x(3x^2 - 16x + 21) = 3x^3 - 16x^2 + 21x
   \]

### Step 2: Expand \( (x^2 - 1)(3x + 1) \)
1. Use the distributive property:
   \[
   (x^2 - 1)(3x + 1) = x^2 \cdot 3x + x^2 \cdot 1 - 1 \cdot 3x - 1 \cdot 1
   \]
   \[
   = 3x^3 + x^2 - 3x - 1
   \]

### Step 3: Substitute and Combine Terms
Now we have:

\[
3x^3 - 16x^2 + 21x - (3x^3 + x^2 - 3x - 1)
\]

Distribute the negative sign:

\[
3x^3 - 16x^2 + 21x - 3x^3 - x^2 + 3x + 1
\]

Combine like terms:

1. \( 3x^3 - 3x^3 = 0 \)
2. \( -16x^2 - x^2 = -17x^2 \)
3. \( 21x + 3x = 24x \)
4. Constant term: \( +1 \)

So, the simplified expression is:

\[
-17x^2 + 24x + 1
\]

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### Related Questions
1. How do you simplify expressions with multiple variables?
2. What is the distributive property in algebra?
3. How do you combine like terms in polynomial expressions?
4. How can we factor a polynomial expression like this one?
5. What are some techniques for simplifying expressions with multiple factors?

### Tip:
When simplifying expressions, handle each part separately before combining them for a clearer result.

Simplify the expression x(3 - x)(7 - 3x) - (x^2 - 1)(3x + 1).

Learn how to simplify the polynomial expression x(3 - x)(7 - 3x) - (x^2 - 1)(3x + 1) with step-by-step expansion and combination of like terms. This problem covers key algebraic concepts, including polynomial expansion and the distributive property. Suitable for students in Grades 9-11.

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