We need to find two numbers that multiply to 64 (the c value of the quadratic) and add up to 16 (the b value of the quadratic).
It helps to list out the factor pairs of 64 and their sums:
color(white){color(black)(
(1,+qquad 64 quad=65),
(2,+qquad32 quad=34),
(4,+qquad16 quad=20),
(8,+qquad8 qquad=16color(red)larr),
(16,+qquad4 qquad=20),
(32,+qquad2 qquad=34),
(64,+qquad1 qquad=65):}
The only factors of 64 that add up to 16 are 8 and 8 (highlighted with a red arrow).
Now, split up the n terms in the quadratic to 8 and 8:
color(white)=n^2+16n+64
=n^2+8n+8n+64
Now, factor the first two terms and the last two terms separately, then combine them:
color(white)=n^2quad+quad8nquad+quad8nquad+quad64
=overbrace(color(red)n*n)+overbrace(color(red)n*8)+overbrace(color(blue)8*n)+overbrace(color(blue)8*8)
=color(red)n(n+8)+color(blue)8(n+8)
=(color(red)n+color(blue)8)(n+8)
=(n+8)^2
That's the most factored it can get. Hope this helped!
