Factorise `x^2-24x-180`

Answer

To factorise the expression \(x^2 - 24x - 180\), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 1\)
- \(b = -24\)
- \(c = -180\)

### Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(ac = 1 \times -180 = -180\) and add to \(b = -24\).

After checking the pairs of factors of \(-180\), we find:
- The numbers \(-30\) and \(6\) satisfy the conditions:
- \(-30 \times 6 = -180\)
- \(-30 + 6 = -24\)

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression by splitting the middle term \(-24x\) into \(-30x + 6x\):
\[ x^2 - 30x + 6x - 180 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 - 30x) + (6x - 180) \]

### Step 5: Factor out the common terms from each group
From the first group \(x^2 - 30x\), we can factor out \(x\):
\[ x(x - 30) \]
From the second group \(6x - 180\), we can factor out \(6\):
\[ 6(x - 30) \]

### Step 6: Combine the factored terms
Now we can combine the factored terms:
\[ x(x - 30) + 6(x - 30) \]
This can be written as:
\[ (x - 30)(x + 6) \]

### Final Result
Thus, the factorised form of the expression \(x^2 - 24x - 180\) is:
\[ (x - 30)(x + 6) \]

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