Answer
The solutions are $2.5$ and $8$.
Work Step by Step
Subtract $5$ to both sides to obtain:
$2(m-3)^2-9(m-3)-5=0$
Let $u=m-3$.
Rewrite the equation above using $u$ to obtain:
$2u^2-9u-5=0$
Factor the trinomial to obtain:
$(2u+1)(u-5)=0$
Use the Zero Factor Property by equating each factor to zero, then solve each equation to obtain:
\begin{array}{ccc}
\\&2u+1=0 &\text{or} & u-5=0
\\&2u=-1 &\text{or} & u=5
\\&u=-\frac{1}{2} &\text{or} & u=5
\end{array}
Since $u=m-3$, then
\begin{array}{ccc}
\\&u=-\frac{1}{2} &\text{or} & u=5
\\&m-3=-\frac{1}{2} &\text{or} & m-3=5
\\&m=-\frac{1}{2}+3 &\text{or} & m=5+3
\\&m=2.5 &\text{or} & m=8
\end{array}
Therefore, the solutions are $2.5$ and $8$.
