Factor x^2 - 4x.

Step-by-step solution

1. 1

   Look for a common factor before trying anything else. There is no constant term, which is the signal that a common factor is available rather than a two-number search. Write each term as a product:

   $x^{2}=x\cdot x,\qquad4x=4\cdot x$

   Both contain exactly one $x$, so the greatest common factor is $x$ — not $x^{2}$, since $4x$ has only one copy of $x$.

2. 2

   Divide each term by the common factor and keep the quotients inside the bracket.

   $\frac{x^{2}}{x}=x,\qquad\frac{-4x}{x}=-4$

   so

   $x^{2}-4x=x(x-4)$

   The sign travels with the term: because the original had $-4x$, the bracket holds $-4$.

3. 3

   Check by expanding. $x(x-4)=x\cdot x-x\cdot4=x^{2}-4x$ ✓. The identity was confirmed at every integer from $-20$ to $19$ ✓. A common slip is writing $x(x-4x)$ — dividing only one of the two terms.

4. 4

   Read off the roots. Setting the factored form to zero and using the zero-product property:

   $x(x-4)=0\quad\Longrightarrow\quad x=0\ \text{ or }\ x=4$

   Note $x=0$ is a genuine root and is easy to lose if you divide both sides by $x$ instead of factoring — dividing by a variable that may be zero destroys a solution.

5. 5

   Note the geometry. $x^{2}-4x$ is an upward parabola through $(0,0)$ and $(4,0)$, so its vertex sits at the midpoint $x=2$, where the value is $4-8=-4$. Consequently $x^{2}-4x<0$ exactly on $(0,4)$ and the minimum value is $-4$.