What on earth does "$r$ is not a root" even mean? Method of Undetermined Coeff Learning ODE now, and using method of Undetermined Coeff
$$y'' +3y' - 7y = t^4 e^t$$
The book said that $r = 1$ is not a root of the characteristic equation. The characteristic eqtn is $r^2 + 3r - 7 = 0$ and the roots are $r = -3 \pm \sqrt{37}$
Where on earth are they getting $r = 1$ from?

$1$ comes from the $e^t$ on the right side.  If it was $e^{kt}$ they would take $r=k$.
