Solve the Algebraic Expression: (x-8)(x+8) + 3x + 17 = -49

To solve this problem, we'll follow these steps:

• Step 1: Simplify the quadratic expression using the difference of squares formula.
• Step 2: Combine like terms and bring all terms to one side to form a standard quadratic equation.
• Step 3: Solve the quadratic equation using factoring or the quadratic formula.

Let's work through each step:
Step 1: The expression $(x-8)(x+8)$ can be expanded using the difference of squares formula: $(x-8)(x+8) = x^2 - 64.$

Substituting this into the equation: $x^2 - 64 + 3x + 17 = -49.$

Step 2: Combine like terms: $x^2 + 3x + 17 - 64 = -49.$

Simplifying further: $x^2 + 3x - 47 = -49.$

Add 49 to both sides to form the standard quadratic equation: $x^2 + 3x + 2 = 0.$

Step 3: Now, solve the quadratic equation. Notice that we can factor it: $x^2 + 3x + 2 = (x + 1)(x + 2) = 0.$

This gives us two possible solutions: $x + 1 = 0 \quad \Rightarrow \quad x = -1,$ $x + 2 = 0 \quad \Rightarrow \quad x = -2.$

Therefore, the solutions to the problem are $x = -1, -2$.