Factor x^2 y - 4xy + 4y.

Step-by-step solution

1. 1

   Look for a common factor across all three terms first. Each term contains exactly one $y$ and no common power of $x$ (the last term has none):

   $x^2y=y\cdot x^2,\qquad -4xy=y\cdot(-4x),\qquad 4y=y\cdot 4$

   So the greatest common factor is $y$.

2. 2

   Factor out the GCF.

   $x^2y-4xy+4y=y\left(x^2-4x+4\right)$

   Always doing this first is what makes the second stage easy: the bracket is now monic.

3. 3

   Recognise the bracket as a perfect square. $x^2=(x)^2$, $4=2^2$, and the middle term satisfies $2\cdot x\cdot 2=4x$ with the required minus sign:

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

4. 4

   Combine both stages.

   $x^2y-4xy+4y=y(x-2)^2$

   The word completely in the instruction means neither stage may be skipped — $y(x^2-4x+4)$ is not yet a complete factorisation.

5. 5

   Check with a substitution. At $x=5$, $y=3$: the original is $75-60+12=27$, and $y(x-2)^2=3\cdot 9=27$ $\checkmark$.