# Factor 20mn+p^2-4m^2-25n^2

20mn+p2-4m2-25n2
Regroup terms.
p2+20mn-4m2-25n2
p2+20(mn)-4m2-25n2
Let u=mn. Substitute u for all occurrences of mn.
p2+20u-4m2-25n2
Factor -1 out of 20u-4m2-25n2.
Move 20u.
p2-4m2-25n2+20u
Factor -1 out of -4m2.
p2-(4m2)-25n2+20u
Factor -1 out of -25n2.
p2-(4m2)-(25n2)+20u
Factor -1 out of 20u.
p2-(4m2)-(25n2)-(-20u)
Factor -1 out of -(4m2)-(25n2).
p2-(4m2+25n2)-(-20u)
Factor -1 out of -(4m2+25n2)-(-20u).
p2-(4m2+25n2-20u)
p2-(4m2+25n2-20u)
Replace all occurrences of u with mn.
p2-(4m2+25n2-20(mn))
Remove parentheses.
p2-(4m2+25n2-20mn)
Factor using the perfect square rule.
Rearrange terms.
p2-(4m2-20mn+25n2)
Rewrite 4m2 as (2m)2.
p2-((2m)2-20mn+25n2)
Rewrite 25n2 as (5n)2.
p2-((2m)2-20mn+(5n)2)
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
20mn=2⋅(2m)⋅(5n)
Rewrite the polynomial.
p2-((2m)2-2⋅(2m)⋅(5n)+(5n)2)
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=2m and b=5n.
p2-(2m-5n)2
p2-(2m-5n)2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=p and b=2m-5n.
(p+2m-5n)(p-(2m-5n))
Simplify.
Apply the distributive property.
(p+2m-5n)(p-(2m)-(-5n))
Multiply 2 by -1.
(p+2m-5n)(p-2m-(-5n))
Multiply -5 by -1.
(p+2m-5n)(p-2m+5n)
(p+2m-5n)(p-2m+5n)
Factor 20mn+p^2-4m^2-25n^2

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