Factorise the polynomial `x^(2)+7x+12` by splitting the middle term .

Answer

To factorise the polynomial \( x^2 + 7x + 12 \) by splitting the middle term, we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \( x^2 + 7x + 12 \).
Here, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 7 \) (coefficient of \( x \))
- \( c = 12 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times 12 = 12 \) and add to \( b = 7 \).

The two numbers that satisfy these conditions are \( 3 \) and \( 4 \):
- \( 3 \times 4 = 12 \)
- \( 3 + 4 = 7 \)

### Step 3: Rewrite the middle term
Now, we can rewrite the polynomial by splitting the middle term \( 7x \) into \( 4x + 3x \):
\[ x^2 + 4x + 3x + 12 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 + 4x) + (3x + 12) \]

### Step 5: Factor out the common factors
Now, we factor out the common factors from each group:
\[ x(x + 4) + 3(x + 4) \]

### Step 6: Factor by grouping
We can now factor out the common binomial factor \( (x + 4) \):
\[ (x + 4)(x + 3) \]

### Final Result
Thus, the factorised form of the polynomial \( x^2 + 7x + 12 \) is:
\[ (x + 4)(x + 3) \]

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