[tex](1,9)[/tex]
1) Since g(x) is displayed in a table, let's find which rule g(x) has by picking two points: (-2,3), (-1,5) let's find the slope between those points:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-3}{-1-(-2)}=\frac{2}{-1+2}=\frac{2}{1}=2[/tex]
So, the slope is 2. Note that in the table we can see the point (0,7) so we can tell the y-intercept is 7. And the rule is:
[tex]g(x)=2x+7[/tex]
2) Now, let's equate f(x) to g(x) and find the x coordinate of the shared point:
[tex]\begin{gathered} 5x^2+x+3=2x+7 \\ 5x^2+x+3-7=2x+7-7 \\ 5x^2+x-4=2x \\ 5x^2-x-4=0 \\ x_=\frac{-\left(-1\right)\pm\sqrt{\left(-1\right)^2-4\cdot\:5\left(-4\right)}}{2\cdot\:5} \\ x_=1,-\frac{4}{5} \end{gathered}[/tex]
The next step is to plug and check. So let's plug x=1 into each original equation. The true solution(s) will yield the same output.
[tex]\begin{gathered} y=2x+7 \\ y=5x^2+x+3 \\ y=2+7\Rightarrow y=9 \\ y=5(1)^2+(1)+3\Rightarrow y=9 \\ \\ y=2(-\frac{4}{5})+7\Rightarrow y=\frac{27}{5} \\ y=5(-\frac{4}{5})^2-\frac{4}{5}+3\Rightarrow y=\frac{27}{5} \end{gathered}[/tex]
Thus, the answer is the point
[tex](-\frac{4}{5},\frac{27}{5}),(1,9)[/tex]
As there is only one answer in the choices we can tell that the answer is:
(1,9)
