(c) f(x)=-3x(x+1)(x-4)^2 Falls to the left and rises to the right C Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right

To determine the behavior of the function \( f(x) = -3x(x+1)(x-4)^2 \) as \( x \) approaches positive and negative infinity, we need to analyze the leading term of the polynomial.<br /><br />First, let's expand the polynomial to identify the leading term:<br /><br />\[ f(x) = -3x(x+1)(x-4)^2 \]<br /><br />Expanding \( (x-4)^2 \):<br /><br />\[ (x-4)^2 = x^2 - 8x + 16 \]<br /><br />Now, substitute this back into the original function:<br /><br />\[ f(x) = -3x(x+1)(x^2 - 8x + 16) \]<br /><br />Next, distribute \( -3x \) through the terms inside the parentheses:<br /><br />\[ f(x) = -3x(x^2 - 8x + 16) + -3x(x + 1) \]<br />\[ f(x) = -3x^3 + 24x^2 - 48x + -3x^2 - 3x \]<br />\[ f(x) = -3x^3 + 21x^2 - 51x \]<br /><br />The leading term is \( -3x^3 \). Since the coefficient of \( x^3 \) is negative, the function falls to the left and rises to the right.<br /><br />Therefore, the correct answer is:<br /><br />**Falls to the left and rises to the right**