I understand that a positive definite matrix by the definition is a symmetric matrix where all eigenvalues are positive. I also know that if $ (x,y) = {x^T}{\cdotp}M{\cdotp}y$ then it is positive definite if ${x^T}{\cdotp}M{\cdotp}x {\geq} 0$ and ${x^T}{\cdotp}M{\cdotp}x = 0$ only if $x=\vec{0}$.
Now I believe I need to go about this by first proving that all eigenvalues of A are positive using $Ax = {\lambda}x$ where x is a real eigenvector and $\lambda$ is a real eigenvalue. After I've done this I believe I need to then answer the question and prove that it is positive definite only if all eigenvalues are positive, but I'm not entirely sure how to do the second part.
That's not quite right. A symmetric real matrix is said to be positive definite if $x^TMx$ is positive whenever $x\neq\vec 0.$ Nothing is said, here, about eigenvalues.
Now, suppose that $\lambda$ is some eigenvalue of $M,$ meaning that there is some vector $x$ with $x\ne\vec 0$ such that $$Mx=\lambda x.$$ What can you say about $x^TMx$? How can you rewrite $x^TMx$? What does this let you say about $\lambda$?