Simplify, giving the answers in factors: (a+1)(a−18)+(a−2)(a+15)

Options

A. `2 (a - 6) (a - 4)`
B. `2(a-6) (a + 4)`
C. `2 (a - 4) (a + 6)`
D. `2 (a + 6) ( a + 4)`

Answer

To simplify the expression \((a+1)(a−18)+(a−2)(a+15)\) and give the answer in factors, we can follow these steps:

### Step 1: Expand the Expression
We start by expanding each term in the expression.

\[ (a+1)(a-18) = a^2 - 18a + a - 18 = a^2 - 17a - 18 \]

\[ (a-2)(a+15) = a^2 + 15a - 2a - 30 = a^2 + 13a - 30 \]

Now, we can combine these two results:

\[ a^2 - 17a - 18 + a^2 + 13a - 30 \]

### Step 2: Combine Like Terms
Now we combine the like terms:

\[ 2a^2 + (-17a + 13a) + (-18 - 30) = 2a^2 - 4a - 48 \]

### Step 3: Factor the Quadratic Expression
Next, we need to factor the quadratic expression \(2a^2 - 4a - 48\).

First, we can factor out the common factor of 2:

\[ 2(a^2 - 2a - 24) \]

Now, we need to factor the quadratic \(a^2 - 2a - 24\). We look for two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(6\) and \(-4\).

Thus, we can factor it as:

\[ a^2 - 2a - 24 = (a - 6)(a + 4) \]

### Step 4: Write the Final Factored Form
Substituting back, we get:

\[ 2(a - 6)(a + 4) \]

### Final Answer
The simplified expression in factors is:

\[ 2(a - 6)(a + 4) \]

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