For what values of $s$ and $t$ is the matrix negative semidefinite? 
Let $$A =
\pmatrix{s&s&0\\
s&s&0\\
0&0&t}$$ where $s, t \in \mathbb R$ are parameters, and let $QA : R_
3 \to R$ be the corresponding quadratic form. Determine for which values of $s$ and $t$, $A$ and $QA$ are negative semidefinite and indefinite.

So far I calculated the product of X transposed AX and gave me that below:
(https://i.stack.imgur.com/9Xd9d.jpg)
I then said that this:
(https://i.stack.imgur.com/JmZUe.jpg)
Is that ok? If so how would I go about saying what values are indefinite.
Many thanks for you help.

Your first step looks okay, you write the quadratic form as $$s(x+y)^2 + tz^2.$$
However, now you need to look for $s$ and $t$ such that for all $x, y, z$ the quadratic form is nonpositive ($\leq 0$). This must be a a condition on $s$ and $t$ that is independent of $x, y, z$. However, since $(x+y)^2>0$ for all $x,y$ and $z^2>0$  for all $z$, it is easy: if $s\leq 0$ and $t\leq 0$ you are sure that $$s(x+y)^2 + tz^2\leq 0 \quad \forall x,y,z.$$
Now what happens if $s$ and $t$ have opposite sign?
