Answer
$x\approx -1.90 $, $x\approx 2.72 $
Work Step by Step
Given \begin{equation}
\sqrt{12 x^2+5 x-8}=7+x.
\end{equation}First square both sides of the radical equation to eliminate the radical sign. Rearrange and solve for $x$.
\begin{equation}
\begin{aligned}
\left( \sqrt{12 x^2+5x-8}\right)^2&= \left( 7+x\right)^2\\
12 x^2+5x-8& =49+14x+x^2\\
11x^2-9x-57& = 0.
\end{aligned}
\end{equation} Use the quadratic formula to find the value(s) of $x$.
\begin{equation}
\begin{aligned}a &= 11\ , b= -9\ ,\ c = -57\\
x & = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
x&=\frac{-(-9) \pm \sqrt{(-9)^2-4 \cdot 3 \cdot(-57)}}{2 \cdot 11} \\
&=\frac{9\pm\sqrt{2589}}{22}\\
\implies x& = \frac{9+\sqrt{2589}}{22}\\
& \approx2.722\\
x& = \frac{9-\sqrt{2589}}{22}\\
& \approx -1.904\\
\end{aligned}
\end{equation}Check.
\begin{equation}
\begin{aligned}
\sqrt{12\cdot (-1.904)^2+5\cdot(-1.904)-8}& \stackrel{?}{=} 5-1.904 \\
5.10& =5.10\quad \textbf{True}\\
\sqrt{12\cdot ( 2.722)^2+5\cdot( 2.722)-8}& \stackrel{?}{=} 5+ 2.722 \\
9.72& =9.72\quad \textbf{True}
\end{aligned}
\end{equation} The solution is $x\approx -1.90 $ and $x\approx 2.72 $
