We present a new method of investigating the so-called quasilinear strongly-damped wave equationsdomains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity φ is less than 6 and f may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case φ ≡ 0 which corresponds to the so-called semi-linear strongly-damped wave equation, our result allows to remove the long-standing growth restriction | f (u)| C (1 + |u| 5 ).