Factorise:
`z ^(2) +19z-150`

Answer

To factorise the expression \( z^2 + 19z - 150 \), we can follow these steps:

### Step 1: Identify coefficients
The expression is in the standard quadratic form \( az^2 + bz + c \), where:
- \( a = 1 \) (coefficient of \( z^2 \))
- \( b = 19 \) (coefficient of \( z \))
- \( c = -150 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ 1 \times (-150) = -150 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-150\) and add up to \(19\). After checking the pairs of factors of \(-150\), we find:
- The numbers \(25\) and \(-6\) satisfy the conditions:
- \(25 \times (-6) = -150\)
- \(25 + (-6) = 19\)

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term using the numbers found:
\[ z^2 + 25z - 6z - 150 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (z^2 + 25z) + (-6z - 150) \]

### Step 6: Factor out the common terms
Now we can factor out the common factors from each group:
1. From the first group \(z^2 + 25z\), we can factor out \(z\):
\[ z(z + 25) \]
2. From the second group \(-6z - 150\), we can factor out \(-6\):
\[ -6(z + 25) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ z(z + 25) - 6(z + 25) \]
This can be written as:
\[ (z - 6)(z + 25) \]

### Final Answer
Thus, the factorised form of \( z^2 + 19z - 150 \) is:
\[ (z - 6)(z + 25) \]

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