## Expanding the Expression (x+1)(x-6)

The expression (x+1)(x-6) represents the product of two binomials. To expand this expression, we can use the **FOIL** method.

**FOIL** stands for:

**F**irst: Multiply the first terms of each binomial.**O**uter: Multiply the outer terms of the binomials.**I**nner: Multiply the inner terms of the binomials.**L**ast: Multiply the last terms of each binomial.

Let's apply this to our expression:

**F:** (x)(x) = x²
**O:** (x)(-6) = -6x
**I:** (1)(x) = x
**L:** (1)(-6) = -6

Now we add all the terms together:

x² - 6x + x - 6

Combining the like terms (-6x + x), we get:

**x² - 5x - 6**

Therefore, the expanded form of (x+1)(x-6) is **x² - 5x - 6**.

### Factoring the Expression

We can also factor the expression x² - 5x - 6 back into its original form (x+1)(x-6). To do this, we need to find two numbers that:

- Multiply to -6 (the constant term)
- Add up to -5 (the coefficient of the x term)

The numbers 1 and -6 satisfy these conditions. Therefore, we can factor the expression as follows:

**x² - 5x - 6 = (x + 1)(x - 6)**

### Uses of the Expression

The expression (x+1)(x-6) and its expanded form, x² - 5x - 6, have many applications in mathematics, particularly in algebra and calculus. Here are some examples:

**Solving quadratic equations:**The expression x² - 5x - 6 can be used to solve quadratic equations by setting it equal to zero and finding the roots.**Graphing quadratic functions:**The expanded form of the expression can be used to graph the quadratic function y = x² - 5x - 6, revealing the shape of the parabola and its key features.**Analyzing polynomial functions:**The expression can be used to study the behavior of polynomial functions, including finding critical points, intervals of increase and decrease, and extrema.

Understanding how to expand and factor expressions like (x+1)(x-6) is crucial for a strong foundation in mathematics and its many applications.