Answer
$\left(-\dfrac{2}{3},-\dfrac{17}{3}\right)$ and $\left(4,41\right)$
Work Step by Step
To find the points of intersection of $f(x)$ and $g(x)$, solve $f(x)=g(x)$:
\begin{align*}
3x^2-7&=10x+1\\
3x^2-10x-7-1&=0\\
0&=3x^2-10x-8
\end{align*}
By Factoring:
$$0=(3x+2)(x-4)$$
Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
\begin{align*}
3x+2 &=0 &\text{ or }& &x -4=0\\
3x&=-2 &\text{ or }& &x=4\\
x&=-\dfrac{2}{3} &\text{ or }& &x =4\\
\end{align*}
To find the $y$-coordinates of the points of intersection, evaluate either of the two functions at $x=-\dfrac{2}{3}$ and $x=4$ to obtain:
$g(-\dfrac{2}{3})=10(-\dfrac{2}{3})+1=-\dfrac{17}{3}$
$g(4)=10(4)+1=41$
Therefore, the points of intersection are:
$\left(-\dfrac{2}{3},-\dfrac{17}{3}\right)$ and $\left(4,41\right)$
