\(x^2+10x+7=0\ \ \ \ |subtract\ 7\ from\ both\ sides\\\\x^2+2x\cdot5=-7\ \ \ |add\ 5^2\ to\ both\ sides\\\\x^2+2x\cdot5+5^2=-7+5^2\ \ \ |use\ a^2+2ab+b^2=(a+b)^2\\\\(x+5)^2=-7+25\\\\(x+5)^2=18\iff x+5=\pm\sqrt{18}\\\\x+5=-\sqrt{18}\ or\ x-5=\sqrt{18}\\\\x+5=-\sqrt{9\cdot2}\ or\ x-5=\sqrt{9\cdot2}\\\\x+5=-\sqrt9\cdot\sqrt2\ or\ x-5=\sqrt9\cdot\sqrt2\\\\x+5=-3\sqrt2\ or\ x-5=3\sqrt2\ \ \ |subtract\ 5\ from\ both\ sides\\\\\boxed{x=-5-3\sqrt2\ or\ x=-5+3\sqrt2}\)
