How to find a number a such that an input 10e^-at to a transfer function does not have a Ke^-(at) term in the output
Let \$\small a=4\$, i.e. use the TF's zero at \$\small s=4\$ to cancel the input signal's \$\frac{1}{s+4}\$ component.

This method of cancellation is not as successful as might first be thought. The danger is, developing a 'dipole' (not an antenna!), which is a pole and zero very close to each other on the s-plane. This is unavoidable since it's almost impossible, in practice, to obtain coincident pole and zero locations. The dipole, thus created, gives rise to a characteristic small amplitude, but persistent 'tail' to the transient response.