Factorise:
`4x ^(2) + 9y^(2) + 12 xy`

Options

A. `=(2x-3y)^(2)`
B. `=(2x+3y)(2x-3y)`
C. `=(2x+3y)^(2)`
D. `=(2x+3y)^(3)`

Answer

To factorise the expression \( 4x^2 + 9y^2 + 12xy \), we can follow these steps:

### Step 1: Rearrange the expression
We can rewrite the expression in a more recognizable form:
\[ 4x^2 + 12xy + 9y^2 \]

### Step 2: Identify the structure
Notice that the expression \( 4x^2 + 12xy + 9y^2 \) resembles the expansion of a perfect square trinomial. A perfect square trinomial takes the form \( (a + b)^2 = a^2 + 2ab + b^2 \).

### Step 3: Identify \( a \) and \( b \)
In our case:
- \( a^2 = 4x^2 \) implies \( a = 2x \)
- \( b^2 = 9y^2 \) implies \( b = 3y \)
- The middle term \( 12xy \) can be expressed as \( 2ab = 2(2x)(3y) = 12xy \)

### Step 4: Write the factorised form
Now that we have identified \( a \) and \( b \), we can write the factorised form:
\[ 4x^2 + 12xy + 9y^2 = (2x + 3y)^2 \]

### Final Answer
Thus, the factorised form of \( 4x^2 + 9y^2 + 12xy \) is:
\[ (2x + 3y)^2 \]

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