To solve the expression \((2x - 1)(3x + 1) + (4x + 1)(x - 5)\), let's break it down step by step:

### Step 1: Expand the first part \((2x - 1)(3x + 1)\)
Using the distributive property (also known as FOIL for binomials):

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

\[
= 6x^2 + 2x - 3x - 1
\]

\[
= 6x^2 - x - 1
\]

### Step 2: Expand the second part \((4x + 1)(x - 5)\)
Using the distributive property again:

\[
(4x + 1)(x - 5) = 4x \cdot x + 4x \cdot (-5) + 1 \cdot x + 1 \cdot (-5)
\]

\[
= 4x^2 - 20x + x - 5
\]

\[
= 4x^2 - 19x - 5
\]

### Step 3: Add the two results together
Now, add the expanded terms from Step 1 and Step 2:

\[
(6x^2 - x - 1) + (4x^2 - 19x - 5)
\]

\[
= (6x^2 + 4x^2) + (-x - 19x) + (-1 - 5)
\]

\[
= 10x^2 - 20x - 6
\]

### Final Answer:
The simplified expression is:

\[
10x^2 - 20x - 6
\]

Would you like more details on any of the steps?

### Related Questions:
1. How do you use the distributive property with binomials?
2. What is the FOIL method and when is it applied?
3. Can you explain the significance of combining like terms in an expression?
4. What are the different ways to factor a quadratic expression?
5. How do you solve quadratic equations using the quadratic formula?

### Tip:
When multiplying binomials, always ensure you distribute each term in the first binomial to each term in the second, which helps avoid errors in expansion.

Learn how to expand and simplify the polynomial expression (2x-1)(3x+1) + (4x+1)(x-5) using the distributive property and combining like terms. This problem introduces key algebraic techniques for polynomial expansion and is suitable for students in Grades 9-10.

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