1. The problem is to determine the end behavior of the given polynomial functions based on their equations. 2. The end behavior of a polynomial function depends on the leading term, which is the term with the highest power of $x$. 3. For the first function $f(x) = 4x^5 + 2x^4 - 7x^3 - 6$, the leading term is $4x^5$. 4. Since the degree 5 is odd and the leading coefficient 4 is positive, the end behavior is: as $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to -\infty$. 5. For the second function $f(x) = 6x^4 + 9x^2 - 6x - 1$, the leading term is $6x^4$. 6. Since the degree 4 is even and the leading coefficient 6 is positive, the end behavior is: as $x \to \pm\infty$, $f(x) \to +\infty$. 7. For the third function $f(x) = -3x(x + 1)(x - 4)^2$, first expand the leading term: - The highest power comes from $x \cdot x \cdot x^2 = x^4$. - The leading coefficient is $-3$ times the coefficients from each factor, which is $-3$. 8. Since the degree 4 is even and the leading coefficient is negative, the end behavior is: as $x \to \pm\infty$, $f(x) \to -\infty$. Final answers: - $4x^5$: end behavior up on right, down on left. - $6x^4$: end behavior up on both sides. - $-3x^4$: end behavior down on both sides.