Answer
Vertex: $(-5/6, 73/12)$
Opens downward
x-intercepts: -2.3, .6
y-intercept: 4
Work Step by Step
$f(x)=-3x^2-5x+4$
$a=-3$, $b=-5$, $c=4$
$x=-b/2a$ is the vertex.
$x=-b/2a$
$x=-(-5)/2(-3)$
$x=5/-6$
$x=-5/6$
$f(x)=-3x^2-5x+4$
$f(-5/6)=-3(-5/6)^2-5(-5/6)+4$
$f(-5/6)=-3*25/36+25/6+4$
$f(-5/6)=-75/36+25/6+4$
$f(-5/6)=-25/12+49/6$
$f(-5/6)=-25/12+98/12$
$f(-5/6)=73/12$
Since the $x^2$ coefficient is negative, the graph opens downward.
$x=0$
$f(x)=-3x^2-5x+4$
$f(0)=-3*0^2-5*0+4$
$f(0)=-3*0-0+4$
$f(0)=0+4$
$f(0)=4$
$y=0$
$f(x)=-3x^2-5x+4$
$0=-3x^2-5x+4$
$a=-3$, $b=-5$, $c=4$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(-5)±\sqrt {(-5)^2-4*-3*4})/2*(-3)$
$x=(5±\sqrt {25+48})/-6)$
$x=(5±\sqrt {73})/-6)$
$\sqrt {73} = 8.5$
$x=(5±\sqrt {73})/-6)$
$x=(5±8.5)/-6$
$x=(5+8.5)/-6$
$x=13.5/-6$
$x=-2.25$
$x=-2.3$
$x=(5-8.5)/-6$
$x=-3.5/-6$
$x=7/12$
$x=.59$
$x=.6$