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?

It may be related to this:
For the equation: $y'' +3y' - 7y = e^{t}$.  Using the method of undetermined coefficients, you would guess that $Ae^t$ is a particular solution of the  equation.  But this wouldn't work if $Ae^t$ were a solution to the homogeneous equation. Then, you'd guess $Ate^t$ for a particular solution (assuming that wasn't a solution to the homogeneous equation, in which case you'd try $t^2e^t$).  To check if $Ae^t$ is a solution to the homogeneous equation, you'd check if  $r=1$ is a solution to the c.e..
In your case, I think, the reason for mentioning  that $r=1$ is not a solution of the c.e., is because that tells you that the guess for your particular solution should contain a term $Ae^t$ (the guess contains other terms because you have $t^4e^t$ on the right hand side).
