PEMDAS question: $F(x) = 3x^2 - x+2$. Find $[f(a)]^2$ How should I go about doing this? $(3a^2-a+2)^2$? 
Thus, $9a^4-a^2+4$

$(3a^2-a+2)^2$ is correct but your expansion of this is incorrect. Remember that:$$(3a^2-a+2)^2=(3a^2-a+2)(3a^2-a+2)=3a^2(3a^2-a+2)-a(3a^2-a+2)+2(3a^2-a+2)$$
You might find this useful: expanding or removing brackets
Here is an example to illustrate the point:$$(3-1+2)^2=(4)^2=16$$If we did this using your result, we would get:$$(3-1+2)^2=3^2-1^2+2^2=9-1+4=12\ne16$$However:$$(3-1+2)^2=(3-1+2)(3-1+2)$$$$=3(3-1+2)-1(3-1+2)+2(3-1+2)$$$$=3(4)-1(4)+2(4)=12-4+8=16$$
More generally:$$(a+b+c)d=ad+bd+cd$$Now image $d=(e+f+g)$, then we would get:$$(a+b+c)d=(a+b+c)(e+f+g)$$$$=ad+bd+cd$$$$=a(e+f+g)+b(e+f+g)+c(e+f+g)$$
