Final answer:
By factoring the quadratic equation x² + 5x + 1 = 5x + 2, we find that it simplifies to x² - 1 = 0, which factors into (x + 1)(x - 1) = 0, yielding the solutions x = -1 and x = 1.
Step-by-step explanation:
To solve for all values of x by factoring the given equation, we first need to rewrite the equation by bringing all terms to one side to form a standard quadratic equation.
The initial equation is x² + 5x + 1 = 5x + 2. We can subtract 5x and 2 from both sides to obtain x² + 5x + 1 - 5x - 2 = 0, simplifying to x² - 1 = 0.
This equation is now recognizably a difference of squares, which factors to (x + 1)(x - 1) = 0. Setting each factor equal to zero gives us the solutions x = -1 and x = 1.