To expand and simplify $(4x - 3)(x + 5)$, we’ll apply the distributive property, often known as FOIL (First, Outer, Inner, Last) for binomials.

1. First Terms: $4x \cdot x = 4x^2$
2. Outer Terms: $4x \cdot 5 = 20x$
3. Inner Terms: $-3 \cdot x = -3x$
4. Last Terms: $-3 \cdot 5 = -15$

Now, combine these results: $4x^2 + 20x - 3x - 15$

Combine like terms ($20x - 3x$): $4x^2 + 17x - 15$

So, the correct answer is: $4x^2 + 17x - 15$

The correct option is the first one: $4x^2 + 17x - 15$.

Would you like a detailed breakdown of each step, or do you have any further questions?

Here are some related questions for further practice:

1. What is the expanded form of $(3x + 2)(2x - 5)$?
2. How would you factor $4x^2 + 17x - 15$ if it were presented in this form?
3. Can you expand and simplify $(5x - 4)(x + 3)$?
4. How do you determine the signs of each term when expanding expressions like this?
5. What changes if one of the binomials has a different sign pattern, such as $(4x + 3)(x - 5)$?

Tip: Always keep track of positive and negative signs carefully when expanding expressions.