`2(x+y)^2-9(x+y)-5`

Answer

To factorize the polynomial \( 2(x+y)^2 - 9(x+y) - 5 \), we can follow these steps:

### Step 1: Substitute \( p \) for \( x + y \)
Let \( p = x + y \). Then, the expression becomes:
\[ 2p^2 - 9p - 5 \]

### Step 2: Factor the quadratic expression
We need to factor \( 2p^2 - 9p - 5 \). To do this, we look for two numbers that multiply to \( 2 \times (-5) = -10 \) and add up to \( -9 \).

The numbers that satisfy this are \( -10 \) and \( 1 \).

### Step 3: Rewrite the quadratic
Now, we can rewrite the quadratic expression using these numbers:
\[ 2p^2 - 10p + 1p - 5 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (2p^2 - 10p) + (1p - 5) \]

### Step 5: Factor by grouping
Now, we factor out the common factors from each group:
\[ 2p(p - 5) + 1(p - 5) \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial \( (p - 5) \):
\[ (p - 5)(2p + 1) \]

### Step 7: Substitute back for \( p \)
Now, we substitute back \( p = x + y \):
\[ (x + y - 5)(2(x + y) + 1) \]

### Final Answer
Thus, the factorization of the polynomial \( 2(x+y)^2 - 9(x+y) - 5 \) is:
\[ (x + y - 5)(2(x + y) + 1) \]

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