Solve (x-3)(x+3) = (x-9)(x+9)+x+5: Factored Form Equation Challenge

To solve this problem, we'll use the following steps:

• Step 1: Recognize and apply the difference of squares formula to both sides of the equation.
• Step 2: Simplify the equation.
• Step 3: Solve for $x$.

Let's go through the solution step-by-step:

Step 1: Simplify each side using the difference of squares formula.

The left-hand side is $(x-3)(x+3) = x^2 - 3^2 = x^2 - 9$.

The right-hand side is $(x-9)(x+9) + x + 5 = (x^2 - 9^2) + x + 5 = x^2 - 81 + x + 5$.

Step 2: Set the simplified expressions equal to each other.

$x^2 - 9 = x^2 - 81 + x + 5$

Step 3: Subtract $x^2$ from both sides to eliminate the quadratic term, simplifying the equation:

$-9 = -81 + x + 5$

Simplify by combining like terms on the right-hand side:

$-9 = x - 76$

Add 76 to both sides to solve for $x$:

$-9 + 76 = x$

$x = 67$

Therefore, the solution to the equation is $\mathbf{x = 67}$.