This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given expression. The expression to factorize is (x^2 - 9)^2 + 16(x^2 - 9) + 64. Step 2: Use substitution to simplify the expression. Let y = x^2 - 9. Substitute y into the expression: y^2 + 16y + 64 Step 3: Factor the quadratic expression in terms of y. This is a perfect square trinomial of the form a^2 + 2ab + b^2 = (a+b)^2. Here, a=y and b=8, since 2(y)(8) = 16y and 8^2 = 64. So, the expression factors to: (y+8)^2 Step 4: Substitute back x^2 - 9 for y. ((x^2 - 9) + 8)^2 Step 5: Simplify the expression inside the parenthesis. (x^2 - 9 + 8)^2 = (x^2 - 1)^2 Step 6: Factor the term inside the parenthesis using the difference of squares formula, a^2 - b^2 = (a-b)(a+b). Here, x^2 - 1 can be written as x^2 - 1^2. So, x^2 - 1 = (x-1)(x+1). Step 7: Substitute this factored form back into the expression. ((x-1)(x+1))^2 Step 8: Apply the exponent to both factors. (x-1)^2 (x+1)^2 The fully factored expression is (x-1)^2 (x+1)^2. (x-1)^2 (x+1)^2 That's 2 down. 3 left today — send the next one.