Linear Algebra - Prove trival solution eigenvalue

Question

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$.

Prove that $(2A+5I)x=0$ has only trival solution.

I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that $-\frac{5}{2}$ is not eigenvalue but how could I conclude that?

Any help will be appreciated.

Answer 1

$$\lambda_1 + \lambda_2 = \textrm{tr}\ A\\ \lambda_1 \lambda_2 = \det A$$

Suppose $\lambda_1 = -\frac{5}{2}$. Then can $\lambda_2$ satisfy those relationships?