### Instruction: The graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$.  Find $a + b$.### Response: To find the vertical asymptotes of a rational function, I need to factor the denominator and set it equal to zero.
That means I need to find the values of $x$ that make $3x^2+5x+2=0$.
This looks like a quadratic equation, so I can try to use the quadratic formula to find the roots.
The quadratic formula says that if I have an equation of the form $ax^2+bx+c=0$, the roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
In this case, $a=3$, $b=5$, and $c=2$, so I can plug those values into the formula and simplify.
I get $x=\frac{-5\pm\sqrt{25-24}}{6}$, which simplifies to $x=\frac{-5\pm 1}{6}$.
This means the roots are $x=\frac{-5+1}{6}=\frac{-2}{3}$ and $x=\frac{-5-1}{6}=-1$.
These are the values of $x$ that make the denominator zero, so they are the vertical asymptotes of the graph.
Therefore, $a=\frac{-2}{3}$ and $b=-1$, and I need to find $a+b$.
To do that, I just add the fractions by finding a common denominator.
I get $a+b=\frac{-2}{3}+(-1)=\frac{-2}{3}-\frac{3}{3}=\frac{-5}{3}$.