Answer
$y^{2}(x+3)(x+1)$
Work Step by Step
$x^{2}y^{2}+4xy^{2}+3y^{2}$
Factor out greatest common factor $y^{2}$
$=y^{2}(x^{2}+4x+3)$
To factor $(x^{2}+4x+3)$ find the numbers whose product is $3$ (constant term ) and the sum is $4$(middle term). Factors are $3$ and $1$ whose product is $3$ and sum is $4$.
$=y^{2}(x^{2}+4x+3)$
$=y^{2}(x^{2}+3x+1x+3)$
Factoring by grouping
$=y^{2}((x^{2}+3x)+(1x+3))$
$=y^{2}(x(x+3)+1(x+3))$
$=y^{2}(x+3)(x+1)$
