Given the function $f(x) = x^2 - 4x$, let's find $f(x + h) - f(x)$.

1. First, substitute $x + h$ into $f(x)$:

$f(x + h) = (x + h)^2 - 4(x + h)$

Expand both terms:

$(x + h)^2 = x^2 + 2xh + h^2$ $-4(x + h) = -4x - 4h$

So,

$f(x + h) = x^2 + 2xh + h^2 - 4x - 4h$

2. Now subtract $f(x)$ from $f(x + h)$:

$f(x + h) - f(x) = \left( x^2 + 2xh + h^2 - 4x - 4h \right) - \left( x^2 - 4x \right)$

Simplify:

$f(x + h) - f(x) = x^2 + 2xh + h^2 - 4x - 4h - x^2 + 4x$

Cancel out $x^2$ and $-4x$:

$f(x + h) - f(x) = 2xh + h^2 - 4h$

Thus, the result is:

$f(x + h) - f(x) = 2xh + h^2 - 4h$

Would you like further details on this? Here are five related questions you could explore:

1. What is the difference quotient $\frac{f(x+h) - f(x)}{h}$ for this function?
2. How does this result relate to the derivative of $f(x)$?
3. What is the general form for $f(x+h) - f(x)$ for a quadratic function?
4. Can you find the second derivative of $f(x) = x^2 - 4x$?
5. How does the value of $h$ affect the expression $f(x+h) - f(x)$?

Tip: When simplifying expressions, always look for common terms to cancel early to reduce complexity.