Factoring and Solving (x+3)^2  16
The expression (x+3)^2  16 represents a quadratic equation in a slightly disguised form. Let's explore how to factor and solve it.
Factoring the Expression

Recognize the Difference of Squares: The expression fits the pattern of a difference of squares: a^2  b^2 = (a + b)(a  b)

Identify 'a' and 'b': In our case, a = (x + 3) and b = 4.

Apply the Formula: (x + 3)^2  16 = [(x + 3) + 4][(x + 3)  4]

Simplify: (x + 3)^2  16 = (x + 7)(x  1)
Solving for x
To find the values of x that make the expression equal to zero, we set the factored expression equal to zero:
(x + 7)(x  1) = 0
This means either (x + 7) = 0 or (x  1) = 0
Solving for x in each case:
 x + 7 = 0 => x = 7
 x  1 = 0 => x = 1
Therefore, the solutions to the equation (x + 3)^2  16 = 0 are x = 7 and x = 1.
Summary
We successfully factored the expression (x + 3)^2  16 into (x + 7)(x  1) by recognizing the difference of squares pattern. This factorization allowed us to solve for the roots of the equation, finding the values of x that make the expression equal to zero.