(a+b+2c+3d)(a-b-2c+3d)=(a-b+2c-3d)(a+b-2c-3d)

To verify the given identity:

\[
(a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d)
\]

we will expand both sides and simplify to check if they are indeed equal.

### Left Side Expansion
Expand \((a + b + 2c + 3d)(a - b - 2c + 3d)\):

\[
\begin{align*}
(a + b + 2c + 3d)(a - b - 2c + 3d) &= a(a - b - 2c + 3d) + b(a - b - 2c + 3d) + 2c(a - b - 2c + 3d) + 3d(a - b - 2c + 3d) \\
&= a^2 - ab - 2ac + 3ad + ba - b^2 - 2bc + 3bd + 2ca - 2cb - 4c^2 + 6cd + 3da - 3db - 6dc + 9d^2 \\
&= a^2 - b^2 - 4c^2 + 9d^2 + (ab - ab) + (3ad + 3da) + (-2ac + 2ca) + (3bd - 3db) + (6cd - 6dc) \\
&= a^2 - b^2 - 4c^2 + 9d^2
\end{align*}
\]

### Right Side Expansion
Expand \((a - b + 2c - 3d)(a + b - 2c - 3d)\):

\[
\begin{align*}
(a - b + 2c - 3d)(a + b - 2c - 3d) &= a(a + b - 2c - 3d) - b(a + b - 2c - 3d) + 2c(a + b - 2c - 3d) - 3d(a + b - 2c - 3d) \\
&= a^2 + ab - 2ac - 3ad - ba - b^2 + 2bc + 3bd + 2ca + 2cb - 4c^2 - 6cd - 3da - 3db + 6dc + 9d^2 \\
&= a^2 - b^2 - 4c^2 + 9d^2 + (ab - ab) + (3ad - 3da) + (-2ac + 2ca) + (3bd - 3db) + (6dc - 6cd) \\
&= a^2 - b^2 - 4c^2 + 9d^2
\end{align*}
\]

Since both sides simplify to \(a^2 - b^2 - 4c^2 + 9d^2\), the given identity holds true.

Thus, 

\[
(a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d)
\]

is indeed correct.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you verify if two polynomial expressions are equivalent?
2. What are some common techniques for expanding and simplifying polynomial expressions?
3. Can you explain the distributive property in algebra?
4. How do you factor complex polynomial expressions?
5. What are some applications of polynomial identities in real-world problems?

**Tip:** When verifying algebraic identities, always expand both sides fully and simplify to ensure they match exactly.

Learn how to verify the polynomial identity (a+b+2c+3d)(a-b-2c+3d)=(a-b+2c-3d)(a+b-2c-3d) with a detailed step-by-step solution. This problem involves expanding and simplifying polynomial expressions. Suitable for high school students studying algebra.

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