a, \(-x^2+2x-7=-\left(x^2-2x+1\right)+1-7=-\left(x-1\right)^2-6\)

Vì \(-\left(x-1\right)^2\le0\Rightarrow-\left(x-1\right)^2-6< 0\) => đpcm

b, \(-x^2-6x-10=-\left(x^2+6x+9\right)+9-10=-\left(x+3\right)^2-1\)

Vì \(-\left(x+3\right)^2\le0\Rightarrow-\left(x+3\right)^2-1< 0\) => đpcm

c, \(-x^2-3x-5=-\left(x^2+3x+\dfrac{9}{4}\right)+\dfrac{9}{4}-5=-\left(x+\dfrac{3}{2}\right)^2-\dfrac{11}{4}\)

Vì \(-\left(x+\dfrac{3}{2}\right)^2\le0\Rightarrow-\left(x+\dfrac{3}{2}\right)^2-\dfrac{11}{4}< 0\) => đpcm

d, \(-x^2+4xy-5y^2-8y-18=-\left(x^2-4xy+5y^2+8y+18\right)\)

\(=-\left[\left(x^2-4xy+4y^2\right)+\left(y^2+8y+16\right)\right]-1\)

\(=-\left[\left(x-2y\right)^2+\left(y+4\right)^2\right]-1=-\left(x-2y\right)^2-\left(y+4\right)^2-1\)

Vì \(-\left(x-2y\right)^2\le0;-\left(y+4\right)^2\le0\Rightarrow-\left(x-2y\right)^2-\left(y+4\right)^2-1< 0\)

=> đpcm