Factorise:
`3y ^(2) + 14y + 8`

Answer

To factorise the expression \(3y^2 + 14y + 8\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ay^2 + by + c\), where:
- \(a = 3\) (coefficient of \(y^2\))
- \(b = 14\) (coefficient of \(y\))
- \(c = 8\) (constant term)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply the coefficient of \(y^2\) (which is \(3\)) by the constant term (which is \(8\)):
\[ 3 \times 8 = 24 \]

### Step 3: Find two numbers that multiply to \(24\) and add to \(14\)
Next, we need to find two numbers that multiply to \(24\) and add up to \(14\).
- The pairs of factors of \(24\) are:
- \(1 \times 24\)
- \(2 \times 12\)
- \(3 \times 8\)
- \(4 \times 6\)

Out of these pairs, \(12\) and \(2\) add up to \(14\):
\[ 12 + 2 = 14 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression \(3y^2 + 14y + 8\) by splitting the middle term \(14y\) into \(12y + 2y\):
\[ 3y^2 + 12y + 2y + 8 \]

### Step 5: Group the terms
Next, we will group the terms:
\[ (3y^2 + 12y) + (2y + 8) \]

### Step 6: Factor out the common factors
Now we will factor out the common factors from each group:
1. From the first group \(3y^2 + 12y\), we can factor out \(3y\):
\[ 3y(y + 4) \]
2. From the second group \(2y + 8\), we can factor out \(2\):
\[ 2(y + 4) \]

### Step 7: Combine the factors
Now we can combine the factored terms:
\[ 3y(y + 4) + 2(y + 4) \]
We can see that \((y + 4)\) is a common factor:
\[ (y + 4)(3y + 2) \]

### Final Answer
Thus, the factorised form of \(3y^2 + 14y + 8\) is:
\[ \boxed{(y + 4)(3y + 2)} \]