Solve the Quadratic Equation: x² + 32x = -256

To solve the quadratic equation $x^2 + 32x = -256$, we will use the method of completing the square.

First, we rewrite the equation by moving all terms to one side:
$x^2 + 32x + 256 = 0$.

Next, we complete the square for the expression $x^2 + 32x$. We want to express it in the form $(x + a)^2$. To do this, take half of the coefficient of $x$ (which is 32), square it, and add and subtract the square inside the expression:
- Half of 32 is 16.
- Squaring 16 gives 256.
- Therefore, $x^2 + 32x = (x + 16)^2 - 256$.

Substitute back into the equation:
$(x + 16)^2 - 256 + 256 = 0$
which simplifies to $(x + 16)^2 = 0$.

To find $x$, solve the equation $(x + 16)^2 = 0$:
Taking the square root of both sides gives $x + 16 = 0$.
Thus, $x = -16$.

Therefore, the solution to the quadratic equation is $x = -16$.