Answer
$(a+b+c)(a- b+c)(a+b-c)(-a+ b+c)$
Work Step by Step
Let $E=4a^{2}c^{2} - (a^{2} - b^{2} + c^{2})^{2}$
First we factor it out as the difference of squares:
$E=(2ac-(a^{2} - b^{2} + c^{2}))(2ac+(a^{2} - b^{2} + c^{2}))$
we can take out the parenthesis of both terms by multiplying by $-1$ on left term and by $1$ on right term:
$E=(2ac- a^{2} + b^{2} - c^{2})(2ac+a^{2} - b^{2} + c^{2})$
Let's take the left term and later we will come to the right one
left term:
$2ac- a^{2} + b^{2} - c^{2}$
we rearrange it to be
$- a^{2}+2ac - c^{2}+ b^{2}$
= $-( a^{2}-2ac + c^{2}- b^{2})$
= $ b^2-(a- c)^{2}$
= $b-a+c)(b+a-c)$
= $(-a+ b+c)(a+b-c)$
right term:
$2ac+a^{2} - b^{2} + c^{2}$
we rearrange it to be
$a^{2} + 2ac + c^{2} - b^{2}$
=$(a+c)^{2} - b^{2}$
=$(a+ c- b)(a+c+b)$
=$(a- b+c)(a+b+c)$
then by combining both terms it would be:
$E=(a+b+c)(a- b+c)(a+b-c)(-a+ b+c)$
