Factorise:
`9m^(2) + 24m + 16`

Answer

To factorise the expression \( 9m^2 + 24m + 16 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 9 \)
- \( b = 24 \)
- \( c = 16 \)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 9 \cdot 16 = 144 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 144 \) and add up to \( 24 \). The numbers are \( 12 \) and \( 12 \) because:
\[ 12 \cdot 12 = 144 \quad \text{and} \quad 12 + 12 = 24 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression by splitting the middle term \( 24m \) into \( 12m + 12m \):
\[ 9m^2 + 12m + 12m + 16 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (9m^2 + 12m) + (12m + 16) \]

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 3m(3m + 4) + 4(3m + 4) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (3m + 4) \):
\[ (3m + 4)(3m + 4) \]
or we can write it as:
\[ (3m + 4)^2 \]

### Final Answer
Thus, the factorised form of \( 9m^2 + 24m + 16 \) is:
\[ (3m + 4)^2 \]

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