x(x-3) +x-5=x+5

Let's solve the equation step by step:

Given: \[ x(x - 3) + x - 5 = x + 5 \]

First, simplify the left side of the equation:

1. Distribute \( x \) in \( x(x - 3) \): \[ x^2 - 3x + x - 5 = x + 5 \]

2. Combine like terms: \[ x^2 - 2x - 5 = x + 5 \]

Next, subtract \( x + 5 \) from both sides: \[ x^2 - 2x - 5 - x - 5 = 0 \] This simplifies to: \[ x^2 - 3x - 10 = 0 \]

Now, we can factor the quadratic equation: We need two numbers that multiply to \(-10\) and add to \(-3\). These numbers are \(-5\) and \(2\).

So, we have: \[ (x - 5)(x + 2) = 0 \]

Now, set each factor to zero:

1. \( x - 5 = 0 \) ⇒ \( x = 5 \)
2. \( x + 2 = 0 \) ⇒ \( x = -2 \)

Thus, the solutions to the equation are: \[ x = 5 \quad \text{and} \quad x = -2 \]