$\displaystyle\lim_{x \to -\infty} \dfrac{\sqrt{x^2+2x}+3x}{\sqrt{4x^2+1-x+7}}\\ =\displaystyle\lim_{x \to -\infty} \dfrac{|x|\sqrt{1+\dfrac{2}{x}}+3x}{|x|\sqrt{4+\dfrac{1}{x}}-x+7}\\ =\displaystyle\lim_{x \to -\infty} \dfrac{-\sqrt{1+\dfrac{2}{x}}+3}{-\sqrt{4+\dfrac{1}{x}}-1+\dfrac{7}{x}}\\ =-\dfrac{2}{3}$