## Expanding the Expression: (a+b)(a^3-3ab-b^2)

This expression involves multiplying two binomials: (a+b) and (a^3 - 3ab - b^2). We can achieve this using the **distributive property** or the **FOIL method**.

### Distributive Property

The distributive property states that for any numbers a, b, and c: a(b+c) = ab + ac

Using this property, we can expand the expression as follows:

(a + b)(a^3 - 3ab - b^2) = a(a^3 - 3ab - b^2) + b(a^3 - 3ab - b^2)

Now, we distribute each term outside the parentheses:

= a^4 - 3a^2b - ab^2 + ba^3 - 3ab^2 - b^3

### Simplifying the Expression

Finally, we combine like terms:

= **a^4 + ba^3 - 3a^2b - 4ab^2 - b^3**

Therefore, the expanded and simplified form of the expression (a+b)(a^3-3ab-b^2) is **a^4 + ba^3 - 3a^2b - 4ab^2 - b^3**.

### Using the FOIL Method

The FOIL method is a shortcut for multiplying two binomials. It stands for **First, Outer, Inner, Last**. It follows these steps:

**First:**Multiply the first terms of each binomial.**Outer:**Multiply the outer terms of the binomials.**Inner:**Multiply the inner terms of the binomials.**Last:**Multiply the last terms of each binomial.

Applying this method to our expression:

**First:**a * a^3 = a^4**Outer:**a * -b^2 = -ab^2**Inner:**b * a^3 = ba^3**Last:**b * -b^2 = -b^3

Combining these terms, we get:

= a^4 - ab^2 + ba^3 - b^3

This result can be further simplified by combining like terms, as done in the previous method, giving us the same final answer: **a^4 + ba^3 - 3a^2b - 4ab^2 - b^3**.

Both methods yield the same result, so choose the method that you find easier to understand and apply.