Quick questions on Graphs of polynomials: end behaviour from the leading term and the multiplicity of each zero

For very large positive or negative $x$, the leading term $a_n x^n$ dwarfs every other term, so the polynomial behaves exactly like that single term. Two things decide the picture: whether the degree $n$ is even or odd, and the sign of the leading coefficient $a_n$.

If a polynomial is fully factored into linear factors (and possibly an irreducible quadratic), its zeroes are read off instantly: each factor $(x - \alpha)$ gives a zero at $x = \alpha$, and the curve meets the x-axis there. An irreducible quadratic factor, one with negative discriminant such as $x^2 + x + 1$, contributes no x-intercept at all, but it still affects the sign of the polynomial between the real zeroes (it is always positive when its leading coefficient is positive).

When a factor is squared the zero is a double root, and the curve touches the axis there and turns back, making a turning point on the axis rather than a crossing. Consider the quartic