Find the total area of the finite domains bounded between the curve $y=x^3−4x$ and the line $x+2y=2$.
I have sketched a graph of the curve and the line and found the points they intersect they $x$-axis and $y$-axis by equating them to 0. Afterwards, I tried to equate both equations together to find the point they intersect and I managed to reduced them down to $2x^3-7x-2=0$. Factorised it and I got $(x-2)(2x^2+4x+1)=0$. Then I tried to find the roots of the equation. Hmm and I got stuck...
The steps that you performed so far are correct. Now, you need to solve for $(x-2)(2x^2+4x+1)=0$. Then, you just need to combine the two roots of the quadratic and the root of $(x-2)$. You can solve for the quadratic using the quadratic formula $x=\begin{align}\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align}$. Next, you could then using the points of intersection as your bounds of integration to determine the areas.