Let's simplify the given expression: 

\[
(3x + 1)(x - 5) - (x - 1)(x + 4)
\]

### Step 1: Expand both products.

1. For \((3x + 1)(x - 5)\):

\[
(3x + 1)(x - 5) = 3x(x - 5) + 1(x - 5)
\]
\[
= 3x^2 - 15x + x - 5 = 3x^2 - 14x - 5
\]

2. For \((x - 1)(x + 4)\):

\[
(x - 1)(x + 4) = x(x + 4) - 1(x + 4)
\]
\[
= x^2 + 4x - x - 4 = x^2 + 3x - 4
\]

### Step 2: Subtract the second expression from the first.

\[
(3x^2 - 14x - 5) - (x^2 + 3x - 4)
\]

Distribute the negative sign to each term in the second expression:

\[
= 3x^2 - 14x - 5 - x^2 - 3x + 4
\]

### Step 3: Combine like terms.

\[
= (3x^2 - x^2) + (-14x - 3x) + (-5 + 4)
\]
\[
= 2x^2 - 17x - 1
\]

Thus, the simplified expression is:

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

Would you like further details on any steps?

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Here are 5 related questions you might find useful:

1. How do you factor a quadratic expression like \( 2x^2 - 17x - 1 \)?
2. What is the difference between expanding and factoring polynomials?
3. Can you solve the quadratic equation \( 2x^2 - 17x - 1 = 0 \)?
4. How does the distributive property apply to polynomial multiplication?
5. How can you use the FOIL method for binomial expansion?

**Tip**: Always be careful with signs when expanding and simplifying expressions, especially when subtracting terms.

Learn how to simplify the polynomial expression (3x + 1)(x - 5) - (x - 1)(x + 4) with detailed steps, including expanding binomials and combining like terms. This tutorial covers key algebraic concepts and is suitable for students in Grades 8-10.

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