## Solving the Equation: (x+3)(x-4) + (x-1)(x+1) = 10

This article will guide you through the steps of solving the equation **(x+3)(x-4) + (x-1)(x+1) = 10**. We'll start by expanding the equation and then use algebraic manipulation to find the values of x that satisfy the equation.

### Expanding the Equation

We'll start by expanding the products on the left-hand side of the equation using the distributive property (or FOIL method):

**(x+3)(x-4)**= x(x-4) + 3(x-4) = x² - 4x + 3x - 12 = x² - x - 12**(x-1)(x+1)**= x(x+1) - 1(x+1) = x² + x - x - 1 = x² - 1

Now, we can substitute these expanded expressions back into the original equation:

**(x² - x - 12) + (x² - 1) = 10**

### Simplifying the Equation

Combining like terms on the left-hand side, we get:

**2x² - x - 13 = 10**

To solve for x, we need to set the equation to zero. Subtracting 10 from both sides gives us:

**2x² - x - 23 = 0**

### Solving the Quadratic Equation

We now have a quadratic equation in the form **ax² + bx + c = 0**. There are a couple of ways to solve this:

**Factoring:**If the quadratic equation can be factored, we can set each factor equal to zero and solve for x. Unfortunately, in this case, the equation does not factor easily.**Quadratic Formula:**The quadratic formula is a more general approach to solving quadratic equations. The formula is:

**x = (-b ± √(b² - 4ac)) / 2a**

In our equation, a = 2, b = -1, and c = -23. Substituting these values into the quadratic formula gives us:

**x = (1 ± √((-1)² - 4 * 2 * -23)) / (2 * 2)**

**x = (1 ± √185) / 4**

Therefore, the solutions to the equation **(x+3)(x-4) + (x-1)(x+1) = 10** are:

**x = (1 + √185) / 4** and **x = (1 - √185) / 4**