Question 999619
 What is the end behavior of the graph of the polynomial function?

recall:

All even-degree polynomials behave, on their ends, like quadratics, and all odd-degree polynomials behave, on their ends, like cubics.

consider are the sign and the degree of the leading term; 
in your case the exponent says that this is a degree 6 polynomial, so the graph will behave roughly like a quadratic: up on both ends or down on both ends
since the sign on the leading coefficient is positive, the graph will be {{{up}}} on{{{ both}}} {{{ends}}}



{{{ f(x) = 3x^6 + 30x^5 + 75x^4}}}=>even-degree polynomial with positive leading coefficient which means the graph will behave roughly like a quadratic an will be {{{up}}} on{{{ both}}} {{{ends}}} 

now we can find solutions to this function:

{{{ f(x) = 3x^4(x^2 + 10x + 25)}}}

{{{ f(x) = 3x^4(x^2 + 5x+5x + 25)}}}

{{{ f(x) = 3x^4((x^2 + 5x)+(5x + 25))}}}

{{{ f(x) = 3x^4(x(x + 5)+5(x + 5))}}}

{{{ f(x) = 3x^4(x + 5)^2}}}

roots:
{{{ 0 = 3x^4(x + 5)^2}}}


if {{{ 0 = 3x^4}}}=>{{{x=4}}} (multiplicity four)
if {{{ 0 = (x + 5)^2}}}=>{{{x=-5}}} (multiplicity two)


{{{ graph( 600, 600, -10, 10, -20, 1500, 3x^6 + 30x^5 + 75x^4) }}}