Expand `(2x - 3x^2)^5`

Answer

To expand the expression \((2x - 3x^2)^5\) using the Binomial Theorem, we will follow these steps:

### Step 1: Identify \(a\), \(b\), and \(n\)
In the expression \((a + b)^n\), we identify:
- \(a = 2x\)
- \(b = -3x^2\)
- \(n = 5\)

### Step 2: Write the Binomial Expansion Formula
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
where \(\binom{n}{r}\) is the binomial coefficient.

### Step 3: Apply the Formula
We will expand \((2x - 3x^2)^5\) using the formula:
\[ (2x - 3x^2)^5 = \sum_{r=0}^{5} \binom{5}{r} (2x)^{5-r} (-3x^2)^r \]

### Step 4: Calculate Each Term
Now we will calculate each term for \(r = 0\) to \(r = 5\):

1. **For \(r = 0\)**:
\[ \binom{5}{0} (2x)^{5} (-3x^2)^{0} = 1 \cdot (2x)^5 \cdot 1 = 32x^5 \]

2. **For \(r = 1\)**:
\[ \binom{5}{1} (2x)^{4} (-3x^2)^{1} = 5 \cdot (2x)^4 \cdot (-3x^2) = 5 \cdot 16x^4 \cdot (-3x^2) = -240x^6 \]

3. **For \(r = 2\)**:
\[ \binom{5}{2} (2x)^{3} (-3x^2)^{2} = 10 \cdot (2x)^3 \cdot 9x^4 = 10 \cdot 8x^3 \cdot 9x^4 = 720x^7 \]

4. **For \(r = 3\)**:
\[ \binom{5}{3} (2x)^{2} (-3x^2)^{3} = 10 \cdot (2x)^2 \cdot (-27x^6) = 10 \cdot 4x^2 \cdot (-27x^6) = -1080x^8 \]

5. **For \(r = 4\)**:
\[ \binom{5}{4} (2x)^{1} (-3x^2)^{4} = 5 \cdot (2x)^{1} \cdot 81x^8 = 5 \cdot 2x \cdot 81x^8 = 810x^9 \]

6. **For \(r = 5\)**:
\[ \binom{5}{5} (2x)^{0} (-3x^2)^{5} = 1 \cdot 1 \cdot (-243x^{10}) = -243x^{10} \]

### Step 5: Combine All Terms
Now we combine all the terms calculated:
\[ (2x - 3x^2)^5 = 32x^5 - 240x^6 + 720x^7 - 1080x^8 + 810x^9 - 243x^{10} \]

### Final Answer
Thus, the expansion of \((2x - 3x^2)^5\) is:
\[ 32x^5 - 240x^6 + 720x^7 - 1080x^8 + 810x^9 - 243x^{10} \]