If $(a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d)$, then $2bc$ is equal to

### Concept & Strategy This problem tests your ability to recognize algebraic structures within large polynomials. By strategically grouping terms, we can transform these complex four-term expressions into simple **Difference of Squares** formats. $$(X + Y)(X - Y) = X^2 - Y^2$$ ### Step-by-Step Solution Given equation: $(a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d)$ **Step 1: Group the terms on the Left Hand Side (LHS)** Notice that $a$ and $3d$ keep their positive signs in both brackets, while $b$ and $2c$ change signs. Let's group them: LHS = $[(a + 3d) + (b + 2c)] \cdot [(a + 3d) - (b + 2c)]$ Using $(X+Y)(X-Y) = X^2 - Y^2$, this becomes: LHS = $(a + 3d)^2 - (b + 2c)^2$ **Step 2: Group the terms on the Right Hand Side (RHS)** Here, $a$ keeps its positive sign, and $-3d$ keeps its negative sign. The terms $b$ and $2c$ alternate signs. Group them carefully: RHS = $[(a - 3d) - (b - 2c)] \cdot [(a - 3d) + (b - 2c)]$ Applying the difference of squares again: RHS = $(a - 3d)^2 - (b - 2c)^2$ **Step 3: Equate and simplify** Equating the simplified LHS and RHS: $(a + 3d)^2 - (b + 2c)^2 = (a - 3d)^2 - (b - 2c)^2$ Rearrange to put similar binomials on the same sides: $(a + 3d)^2 - (a - 3d)^2 = (b + 2c)^2 - (b - 2c)^2$ We use the algebraic identity: $(X + Y)^2 - (X - Y)^2 = 4XY$. Apply this to both sides: Left side: $4(a)(3d) = 12ad$ Right side: $4(b)(2c) = 8bc$ So, $12ad = 8bc$. Divide by 4: $3ad = 2bc$ The question asks for the value of $2bc$. ### Exam Strategy & Shortcut **Sign Grouping Method:** In massive expansion problems, immediately look for which variables flip signs. On the left: $(b+2c)$ flips to $-(b+2c)$. On the right: $(b-2c)$ flips to $-(b-2c)$. This immediately tells you the structure is squares of $(a \pm 3d)$ minus squares of $(b \pm 2c)$. Jumping straight to $(a+3d)^2 - (a-3d)^2 = (b+2c)^2 - (b-2c)^2$ saves enormous amounts of time. ### Common Pitfall Attempting to multiply out the brackets term-by-term. Expanding a 4x4 polynomial creates 16 terms on the left and 16 terms on the right. While they will eventually cancel out, it is guaranteed to consume 5+ minutes and almost always results in a sign error. ### Final Answer Therefore, the correct answer is **$3ad$**.