Having trouble helping my daughter solve the calculus problem that requires finding tangent points between a cubic function and a line

Question

determine any and all locations (x-values) where the cubic function $y=x^3-5x^2 +x$ has tangent lines that are parallel to the line $y = -2x + 8$

Answer 1

Consider $y=-2x+q$. For all $q$, that is a line parallel given to yours. Find the derivative of your cubic function $f'(x)=3x^2-10x+1$ and solve $f'(x)=-2$. You'll find that the solutions are $x_1=-3$ and $x_2=-\frac{1}{3}$. So in these two points the line tangent to your function is parallel to thr given line.

Answer 2

Hint: $$f’(x) = 3x^2-10x+1$$

The tangent lines should be parallel to $y = -2x+8$. Thus, their slopes must also be $-2$. Set the derivative equal to the slope.

$$\implies -2 = 3x^2-10x+1$$

Solving this shouldn’t be hard now.