Factorise:
`3m^(2) + 24m + 36`

Answer

To factorise the expression \(3m^2 + 24m + 36\), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is \(3m^2 + 24m + 36\).
- Coefficient of \(m^2\) (let's call it \(a\)) = 3
- Coefficient of \(m\) (let's call it \(b\)) = 24
- Constant term (let's call it \(c\)) = 36

### Step 2: Multiply \(a\) and \(c\)
Now, we multiply the coefficient of \(m^2\) and the constant term:
\[ a \times c = 3 \times 36 = 108 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
Next, we need to find two numbers that multiply to 108 and add up to 24. The numbers are 18 and 6, since:
\[ 18 \times 6 = 108 \quad \text{and} \quad 18 + 6 = 24 \]

### Step 4: Rewrite the middle term
We can now rewrite the expression by splitting the middle term \(24m\) into \(18m + 6m\):
\[ 3m^2 + 18m + 6m + 36 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (3m^2 + 18m) + (6m + 36) \]

### Step 6: Factor out the common factors
Now, we factor out the common factors from each group:
1. From the first group \(3m^2 + 18m\), we can factor out \(3m\):
\[ 3m(m + 6) \]
2. From the second group \(6m + 36\), we can factor out \(6\):
\[ 6(m + 6) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ 3m(m + 6) + 6(m + 6) \]
This can be factored further since \(m + 6\) is common in both terms:
\[ (m + 6)(3m + 6) \]

### Step 8: Simplify the expression
Finally, we can simplify \(3m + 6\) by factoring out a 3:
\[ 3(m + 2) \]
Thus, the complete factorization of the expression is:
\[ 3(m + 6)(m + 2) \]

### Final Answer
The factorised form of \(3m^2 + 24m + 36\) is:
\[ 3(m + 6)(m + 2) \]

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