Answer
Equation of tangent: $$32x-y-47=0$$
Work Step by Step
Given curve, $y=x^4+1$. The tangent line to this curve is parallel to $32x-y=15$, which means that the slope of the tangent line is $32$ (the slope of the line it is parallel to).
Differentiating $y$ with respect to $x$,
$$\frac{dy}{dx} = 4x^3$$This slope must be equal to $32$, that is, the slope of the required tangent.
$$4x^3=32\implies x^3=8\implies x=2.$$Thus, the tangent touches the curve $y$ at $(2, y(2))\equiv(2,17)$. Thus, using point-slope form, the equation of the tangent is,
$$y-17=32(x-2)=32x-64$$$$32x-y-47=0$$
