\(a^2-b^2=4c^2\Rightarrow16a^2-16b^2=64c^2\)
\(\left(5a-3b+8c\right)\left(5a-3b-8c\right)=\left(5a-3b\right)^2-64c^2\)
\(=\left(5a-3b\right)^2-\left(16a^2-16b^2\right)\)
\(=25a^2-30ab+9b^2-16a^2+16b^2\)
\(=9a^2-30ab+25b^2\)
\(=\left(3a-5b\right)^2\)
(5a-3b+8c)(5a-3b-8c)
=[(5a-3b)+8c)][(5a-3b)-8c]
=(5a-3b)^2-64c^2
Vì a2-b2=4c^2 =>64c^2=16a^2-16b^2
Thay vào biểu thức trên,ta có:
(5a-3b+8c)(5a-3b-8c)
=25a^2-30ab+9a^2-16a^2+16b^2
=25b^2-30ab+9a^2
=(3a-5b)^2 (đpcm)