Factorise the following :
`5m^(2)-8m-4`

Answer

To factorise the expression \(5m^2 - 8m - 4\), we will use the method of middle term splitting. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The expression is in the form \(ax^2 + bx + c\), where:
- \(a = 5\)
- \(b = -8\)
- \(c = -4\)

### Step 2: Multiply \(a\) and \(c\)
Calculate the product of \(a\) and \(c\):
\[ a \cdot c = 5 \cdot (-4) = -20 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-20\) and add to \(-8\). The numbers that satisfy this condition are \(-10\) and \(2\) because:
\[ -10 \cdot 2 = -20 \quad \text{and} \quad -10 + 2 = -8 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term:
\[ 5m^2 - 10m + 2m - 4 \]

### Step 5: Group the terms
Next, we will group the terms:
\[ (5m^2 - 10m) + (2m - 4) \]

### Step 6: Factor out the common factors from each group
From the first group \(5m^2 - 10m\), we can factor out \(5m\):
\[ 5m(m - 2) \]
From the second group \(2m - 4\), we can factor out \(2\):
\[ 2(m - 2) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ 5m(m - 2) + 2(m - 2) \]
Now, we can factor out the common term \((m - 2)\):
\[ (m - 2)(5m + 2) \]

### Final Answer
Thus, the factorised form of \(5m^2 - 8m - 4\) is:
\[ (m - 2)(5m + 2) \]

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