Factorise:
`d ^(2) - d - 90`

Answer

To factorize the expression \( d^2 - d - 90 \), we will use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The expression is in the standard form of a quadratic equation \( ax^2 + bx + c \). Here, we have:
- \( a = 1 \) (coefficient of \( d^2 \))
- \( b = -1 \) (coefficient of \( d \))
- \( c = -90 \)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot (-90) = -90 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-90\) and add to \(-1\). After checking the pairs of factors of \(-90\), we find:
- The numbers \( 9 \) and \(-10\) work because:
\[ 9 \times (-10) = -90 \quad \text{and} \quad 9 + (-10) = -1 \]

### Step 4: Rewrite the middle term
We can rewrite the expression \( d^2 - d - 90 \) using the numbers found:
\[ d^2 + 9d - 10d - 90 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (d^2 + 9d) + (-10d - 90) \]

### Step 6: Factor by grouping
Next, we factor out the common factors from each group:
\[ d(d + 9) - 10(d + 9) \]

### Step 7: Factor out the common binomial
Now, we can factor out the common binomial \( (d + 9) \):
\[ (d + 9)(d - 10) \]

### Final Result
Thus, the factorization of the expression \( d^2 - d - 90 \) is:
\[ (d - 10)(d + 9) \]