Factor -x4y + 6x3y - 5x2y completely.

Step-by-step solution

1. 1

   Take out the greatest common monomial factor. Every term contains at least $x^2$ and one $y$, and the numeric coefficients $-1,6,-5$ share only $1$:

   $-x^4y+6x^3y-5x^2y=x^2y\left(-x^2+6x-5\right)$

   Use the lowest power of each variable: $x^2$, not $x^4$.

2. 2

   Factor the minus out of the trinomial. A negative leading coefficient makes the usual pattern hard to see, so extract it:

   $-x^2+6x-5=-\left(x^2-6x+5\right)$

3. 3

   Factor the monic trinomial. Two numbers with product $+5$ and sum $-6$: both negative, namely $-1$ and $-5$:

   $x^2-6x+5=(x-1)(x-5)$

4. 4

   Reassemble, carrying the minus sign.

   $x^2y\left[-(x-1)(x-5)\right]=-x^2y(x-1)(x-5)$

   An equally valid form moves the sign inside one bracket: $x^2y(1-x)(x-5)$.

5. 5

   Expand back and test numerically. $-x^2y(x^2-6x+5)=-x^4y+6x^3y-5x^2y$ ✓. At $x=2$, $y=1$: the original is $-16+48-20=12$, and $-4(1)(2-1)(2-5)=-4(1)(-3)=12$ ✓. Nothing factors further, so this is complete.