To expand \((P + 2q)^3\), we will use the binomial expansion formula:

\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]

In this case, \( a = P \) and \( b = 2q \). Applying the formula:

\[
(P + 2q)^3 = P^3 + 3P^2(2q) + 3P(2q)^2 + (2q)^3
\]

Now, let's simplify each term:

1. \( P^3 \) stays as \( P^3 \).
2. \( 3P^2(2q) = 6P^2q \).
3. \( 3P(2q)^2 = 3P(4q^2) = 12Pq^2 \).
4. \( (2q)^3 = 8q^3 \).

So, the expanded form is:

\[
P^3 + 6P^2q + 12Pq^2 + 8q^3
\]

Would you like further details or clarification on this process? Here are some related questions to further your understanding:

1. What is the general binomial expansion formula for any power \( n \)?
2. How would you expand \( (P + 3q)^4 \)?
3. Can you expand \( (x + 5y)^2 \) and explain the steps?
4. How does the binomial theorem help in approximations?
5. What is the relationship between binomial expansion and Pascal's triangle?

**Tip:** When expanding binomials, recognizing symmetry in coefficients (like those found in Pascal’s triangle) can simplify the process.

Learn how to expand and simplify the expression (P + 2q)^3 using the binomial expansion formula. This detailed solution explains each step, from applying the binomial theorem to simplifying the result. Suitable for students in Grades 10-12.

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