Factorise:
`3 + 23z - 8z^(2)`

Answer

To factorise the expression \(3 + 23z - 8z^2\), we can follow these steps:

### Step 1: Rearrange the Expression
Rearrange the expression in standard form:
\[ -8z^2 + 23z + 3 \]

### Step 2: Factor Out the Negative Sign
Factor out the negative sign from the expression:
\[ -(8z^2 - 23z - 3) \]

### Step 3: Identify Coefficients
Identify the coefficients for the quadratic \(ax^2 + bx + c\):
- \(a = 8\)
- \(b = -23\)
- \(c = -3\)

### Step 4: Calculate the Product \(ac\)
Calculate the product \(ac\):
\[ ac = 8 \times (-3) = -24 \]

### Step 5: Find Two Numbers that Multiply to \(ac\) and Add to \(b\)
We need to find two numbers that multiply to \(-24\) and add to \(-23\). The numbers are \(-24\) and \(1\):
- \(-24 + 1 = -23\)
- \(-24 \times 1 = -24\)

### Step 6: Split the Middle Term
Rewrite the expression by splitting the middle term using the numbers found:
\[ -8z^2 - 24z + z - 3 \]

### Step 7: Group the Terms
Group the terms into two pairs:
\[ (-8z^2 - 24z) + (z - 3) \]

### Step 8: Factor by Grouping
Factor out the common factors from each group:
\[ -8z(z + 3) + 1(z - 3) \]

### Step 9: Factor Out the Common Binomial
Now, we can factor out the common binomial \((z + 3)\):
\[ -(8z + 1)(z - 3) \]

### Final Result
Thus, the factorised form of the expression \(3 + 23z - 8z^2\) is:
\[ -(8z + 1)(z - 3) \]