Factorise : (iii) `1-3a - 28 a^2`

Answer

To factorise the expression \(1 - 3a - 28a^2\), we can follow these steps:

### Step 1: Rewrite the expression
Start with the expression:
\[ 1 - 3a - 28a^2 \]
To make the factorization easier, we can multiply the entire expression by \(-1\):
\[ - (3a + 28a^2 - 1) \]
This gives us:
\[ 28a^2 + 3a - 1 \]

### Step 2: Identify the product and sum
We need to find two numbers that multiply to give the product of the coefficient of \(a^2\) (which is \(28\)) and the constant term (which is \(-1\)). Thus, we need two numbers that multiply to:
\[ 28 \times (-1) = -28 \]
and add up to the coefficient of \(a\) (which is \(3\)).

### Step 3: Find the factors
The pairs of factors of \(-28\) are:
- \(1\) and \(-28\)
- \(-1\) and \(28\)
- \(2\) and \(-14\)
- \(-2\) and \(14\)
- \(4\) and \(-7\)
- \(-4\) and \(7\)

Among these, we find that:
\[ 7 + (-4) = 3 \]
and
\[ 7 \times (-4) = -28 \]
So, the two numbers we are looking for are \(7\) and \(-4\).

### Step 4: Rewrite the middle term
Now we can rewrite the expression \(28a^2 + 3a - 1\) using these factors:
\[ 28a^2 + 7a - 4a - 1 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (28a^2 + 7a) + (-4a - 1) \]

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 7a(4a + 1) - 1(4a + 1) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial factor \((4a + 1)\):
\[ (4a + 1)(7a - 1) \]

### Final Step: Write the final factorised form
Thus, the factorised form of the expression \(1 - 3a - 28a^2\) is:
\[ -(4a + 1)(7a - 1) \]