Factorise : `4m^(3)n^(2) +12 m^(2)n^(2)+18 m^(4)n^(3)`

Answer

To factorise the expression \( 4m^3n^2 + 12m^2n^2 + 18m^4n^3 \), 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 \(4\), \(12\), and \(18\). The greatest common factor (GCF) of these numbers is \(2\).
- For the variable \(m\), the lowest power present in all terms is \(m^2\).
- For the variable \(n\), the lowest power present is \(n^2\).

Thus, the common factor for the entire expression is \(2m^2n^2\).

### Step 2: Factor out the common factor
Next, we will factor out \(2m^2n^2\) from each term in the expression.

\[ 4m^3n^2 = 2m^2n^2 \cdot (2mn) \]
\[ 12m^2n^2 = 2m^2n^2 \cdot (6) \]
\[ 18m^4n^3 = 2m^2n^2 \cdot (9m^2n) \]

Now, we can rewrite the expression as:

\[ 4m^3n^2 + 12m^2n^2 + 18m^4n^3 = 2m^2n^2(2mn + 6 + 9m^2n) \]

### Step 3: Write the final factored form
The final factored form of the expression is:

\[ 2m^2n^2(2mn + 6 + 9m^2n) \]

### Summary of the solution:
The expression \( 4m^3n^2 + 12m^2n^2 + 18m^4n^3 \) can be factorised as \( 2m^2n^2(2mn + 6 + 9m^2n) \).

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