If we square the sum of x and another number, we get

\begin{align} \qquad (x + a)^2 &= (x + a)(x + a) \\ &= x^2 + \pink{2a}x + a^2\end{align}

The number multiplying the x term is 2 times the number that was added inside the square.

In the problem we're trying to solve, the number multiplying the x term is B, so when we rewrite the expression, the number added inside the square will be half of B. (Which is B2)

\qquad (x + B2)^2 + b

How can we find the value of b?

If we multiply out the square in this expression, we get

\begin{align} \qquad (x + B2)^2 + b &= (x + B2)(x + B2) + b \\ &= x^2 + Bx + D + b \end{align}

This looks just like the given expression if

\qquad D + b = C

So b must be C- D.

POLY_TEXT can be rewritten as:

\qquad (x + B2)^2 + C - D