Factor out completely: 4a^2c^2-(a^2-b^2 c^2)^2

The expression 4a^2c^2 - (a^2-b^2+c^2)^2 has to be factored.

4a^2c^2 - (a^2 - b^2 + c^2)^2

=> (2ac)^2 - (a^2 - b^2 + c^2)^2

=> (2ac - a^2 + b^2 - c^2)(2ac + a^2 - b^2 + c^2)

=> (b^2 - (a^2 - 2ac + c^2))((a^2 + 2ac + c^2) - b^2)

=> (b^2 - (a - c)^2)((a + c)^2 - b^2)

=> (b - a + c)(b + a - c)(a + b + c)(a - b + c)

The factorized form of 4a^2c^2 - (a^2-b^2+c^2)^2 is (b - a + c)(b + a - c)(a + b + c)(a - b + c)

so the possibilities of the measurement of the side of the triangle are either it is longest side or not the longest side

when it is the longest side

s < 25 + 35s < 60

if it is not the longest side

35 < 25 + s10 < sso 10 < s < 60so the perimeter70 < p < 130