Factorise : (i) `10a^(4) x^(2) - 15a^(6) x^(4) + 20 a^(7) x^(5)`

Answer

To factorise the expression \(10a^{4}x^{2} - 15a^{6}x^{4} + 20a^{7}x^{5}\), we will follow these steps:

### Step 1: Identify the common factors
First, we need to identify the common factors in all the terms of the expression.

The coefficients are 10, -15, and 20. The greatest common divisor (GCD) of these numbers is 5.

Next, we look at the variable parts:
- For \(a\): The lowest power is \(a^{4}\).
- For \(x\): The lowest power is \(x^{2}\).

Thus, the common factor for the entire expression is \(5a^{4}x^{2}\).

### Step 2: Factor out the common factor
Now, we will factor out \(5a^{4}x^{2}\) from each term in the expression:

\[ 10a^{4}x^{2} - 15a^{6}x^{4} + 20a^{7}x^{5} = 5a^{4}x^{2} \left( \frac{10a^{4}x^{2}}{5a^{4}x^{2}} - \frac{15a^{6}x^{4}}{5a^{4}x^{2}} + \frac{20a^{7}x^{5}}{5a^{4}x^{2}} \right) \]

Calculating each term inside the parentheses:
- The first term: \(\frac{10a^{4}x^{2}}{5a^{4}x^{2}} = 2\)
- The second term: \(\frac{-15a^{6}x^{4}}{5a^{4}x^{2}} = -3a^{2}x^{2}\)
- The third term: \(\frac{20a^{7}x^{5}}{5a^{4}x^{2}} = 4a^{3}x^{3}\)

Putting it all together, we have:

\[ 10a^{4}x^{2} - 15a^{6}x^{4} + 20a^{7}x^{5} = 5a^{4}x^{2} (2 - 3a^{2}x^{2} + 4a^{3}x^{3}) \]

### Step 3: Write the final factorised form
Thus, the final factorised form of the expression is:

\[ 5a^{4}x^{2} (2 - 3a^{2}x^{2} + 4a^{3}x^{3}) \]