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The given image shows a question asking for the coefficients of \(x^2\) and \(x^3\) in the expansion of \((2x - 1)^{30}\).

We will use the **Binomial Theorem** to expand this expression. The Binomial Theorem states:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

For the given expression \((2x - 1)^{30}\):
- \(a = 2x\)
- \(b = -1\)
- \(n = 30\)

The general term in the expansion is:

\[
T_k = \binom{30}{k} (2x)^{30-k} (-1)^k
\]

This term simplifies to:

\[
T_k = \binom{30}{k} 2^{30-k} x^{30-k} (-1)^k
\]

### Finding the coefficient of \(x^2\):
We need to find the term where the power of \(x\) is 2. That means \(30 - k = 2\), so \(k = 28\). Substituting \(k = 28\) into the general term:

\[
T_{28} = \binom{30}{28} 2^2 x^2 (-1)^{28}
\]

Simplifying:

\[
T_{28} = \binom{30}{28} 4 x^2
\]

Since \(\binom{30}{28} = \binom{30}{2} = \frac{30 \times 29}{2} = 435\), the coefficient of \(x^2\) is:

\[
4 \times 435 = 1740
\]

### Finding the coefficient of \(x^3\):
We need to find the term where the power of \(x\) is 3. That means \(30 - k = 3\), so \(k = 27\). Substituting \(k = 27\) into the general term:

\[
T_{27} = \binom{30}{27} 2^3 x^3 (-1)^{27}
\]

Simplifying:

\[
T_{27} = \binom{30}{27} 8 x^3 (-1)
\]

Since \(\binom{30}{27} = \binom{30}{3} = \frac{30 \times 29 \times 28}{6} = 4060\), the coefficient of \(x^3\) is:

\[
-8 \times 4060 = -32,480
\]

### Final Answers:
- The coefficient of \(x^2\) is **1740**.
- The coefficient of \(x^3\) is **-32,480**.

Would you like further details or have any questions about this expansion?

Here are five related questions to explore:
1. How do binomial coefficients change when the exponent increases?
2. Can you find the coefficient of \(x^4\) in the expansion of \((2x - 1)^{30}\)?
3. What is the sum of all the coefficients in the expansion of \((2x - 1)^{30}\)?
4. How would the expansion differ if the expression were \((2x + 1)^{30}\) instead?
5. What is the general form of the binomial theorem for negative exponents?

**Tip**: The binomial coefficients can be quickly calculated using Pascal's triangle or the formula for combinations \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

In the expansion of (2x - 1)^30, find the coefficients of x^2 and x^3.

Learn how to calculate the coefficients of x^2 and x^3 in the binomial expansion of (2x - 1)^30 using the binomial theorem and combinatorics. This solution involves key algebraic concepts like polynomial expansion and the use of combinations.

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