Factor 4x^2y + 12xy + 9y.

Step-by-step solution

1. 1

   Always look for a common factor first. Every term contains exactly one $y$ (and the coefficients $4,12,9$ share no factor beyond $1$), so $y$ is the greatest common factor:

   $4x^2y+12xy+9y=y\left(4x^2+12x+9\right)$

   Skipping this step and trying to factor the trinomial in two variables directly makes the problem look far harder than it is.

2. 2

   Test whether the bracket is a perfect square. A trinomial $A^2+2AB+B^2$ needs its outer terms to be squares and its middle term to be twice the product of their roots. Here $4x^2=(2x)^2$ and $9=3^2$, so check the middle:

   $2\cdot(2x)\cdot3=12x\ \checkmark$

   It matches exactly.

3. 3

   Write the square.

   $4x^2+12x+9=(2x+3)^2$

   Both signs are positive, so the binomial is a sum rather than a difference. (Confirming independently: the discriminant is $144-4(4)(9)=144-144=0$, the signature of a repeated root.)

4. 4

   Combine with the common factor.

   $4x^2y+12xy+9y=y(2x+3)^2$

5. 5

   Read off what the form tells you. Since $(2x+3)^2\ge0$ always, the expression has the same sign as $y$, and it vanishes exactly when $y=0$ or $x=-\tfrac32$. The double root at $x=-\tfrac32$ means the parabola (for fixed $y>0$) touches the axis there rather than crossing it.

6. 6

   Verify numerically. Comparing the original with $y(2x+3)^2$ at $50$ random pairs $(x,y)$ drawn from $[-5,5]$ gives agreement to machine precision at every point ✓. Spot check at $x=1,y=2$: $8+24+18=50$, and $2(5)^2=50$ ✓.