\( \frac{x^{2}+5 x}{2}+\frac{12}{x^{2}+5 x}=-5 \) let \( x^{2}+5 x=a \)

Solve the equation \( \frac{x^{2}+5x}{2}+\frac{12}{x^{2}+5x}=-5 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x^{2}+5x}{2}+\frac{12}{x^{2}+5x}=-5\) - step1: Find the domain: \(\frac{x^{2}+5x}{2}+\frac{12}{x^{2}+5x}=-5,x \in \left(-\infty,-5\right)\cup \left(-5,0\right)\cup \left(0,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{x^{2}+5x}{2}+\frac{12}{x^{2}+5x}\right)\times 2\left(x^{2}+5x\right)=-5\times 2\left(x^{2}+5x\right)\) - step3: Simplify the equation: \(x^{4}+10x^{3}+25x^{2}+24=-10x^{2}-50x\) - step4: Move the expression to the left side: \(x^{4}+10x^{3}+25x^{2}+24-\left(-10x^{2}-50x\right)=0\) - step5: Calculate: \(x^{4}+10x^{3}+35x^{2}+24+50x=0\) - step6: Factor the expression: \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x+1=0\\&x+2=0\\&x+3=0\\&x+4=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=-1\\&x=-2\\&x=-3\\&x=-4\end{align}\) - step9: Check if the solution is in the defined range: \(\begin{align}&x=-1\\&x=-2\\&x=-3\\&x=-4\end{align},x \in \left(-\infty,-5\right)\cup \left(-5,0\right)\cup \left(0,+\infty\right)\) - step10: Find the intersection: \(\begin{align}&x=-1\\&x=-2\\&x=-3\\&x=-4\end{align}\) - step11: Rewrite: \(x_{1}=-4,x_{2}=-3,x_{3}=-2,x_{4}=-1\) The solutions to the equation \( \frac{x^{2}+5x}{2}+\frac{12}{x^{2}+5x}=-5 \) are \( x=-4, -3, -2, -1 \).