The following definition is given in Scahum's Ouline of Linear Algebra:
\begin{array}{l}{\text { DEFINTION A: A mapping } q: V \rightarrow K \text { is a quadratic form if } q(v)=f(v, v) \text { for some symmetric }} \\ {\text { bilinear form } f \text { on } V \text { . }} \\ {\text {If } 1+1 \neq 0\text{ in } K,}\\{\text {then the bilinear form } f \text { can be obtained from the quadratic form } q \text { by the following } polar form }\\{\text{ of }f\text{:}} \\ {\qquad f(u, v)=\frac{1}{2}[q(u+v)-q(u)-q(v)]}\end{array}
Where does the last equation appear from, and what does the book mean by polar form?
You have to compute $q(u+v)$. Using bilinearity, $$ q(u+v)=f(u+v,u+v)=f(u,u)+f(u,v)+f(v,u)+f(v,v)=f(u,v)+f(v,u)+q(u)+q(v). $$ Now, if your field has characteristic $>2$, $f(u,v)+f(v,u)=2f(u,v)$ and you just solve for $f(u,v)$. This identity is called the polarization identity, and it is related to the concept of polar in projective geometry.