`(a ^(2) + 2a) ^2 + 12 ( a ^(2) + 2a) - 45 ` can be expressed as :

Options

A. `(a-1)(a+3)(a ^(2) + 2a+ 15)`
B. `(a+ 1) (a + 3)(a ^(2) + 2a + 15)`
C. `(a +1) (a -3) (a^(2) + 2a + 15)`
D. `(a-1) (a-3) (a ^(2) + 2a + 15)`

Answer

To express the given expression \((a^2 + 2a)^2 + 12(a^2 + 2a) - 45\), we can follow these steps:

### Step 1: Substitute \(k\)
Let \(k = a^2 + 2a\). This simplifies our expression.

So, we rewrite the expression as:
\[ k^2 + 12k - 45 \]

### Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression \(k^2 + 12k - 45\).

To factor this, we look for two numbers that multiply to \(-45\) (the constant term) and add up to \(12\) (the coefficient of \(k\)). The numbers that satisfy this condition are \(15\) and \(-3\).

Thus, we can rewrite the expression as:
\[ k^2 + 15k - 3k - 45 \]

### Step 3: Group the Terms
Now, we group the terms:
\[ (k^2 + 15k) + (-3k - 45) \]

### Step 4: Factor by Grouping
Now we factor each group:
\[ k(k + 15) - 3(k + 15) \]

### Step 5: Factor Out the Common Factor
Now, we can factor out the common factor \((k + 15)\):
\[ (k + 15)(k - 3) \]

### Step 6: Substitute Back for \(k\)
Finally, we substitute back \(k = a^2 + 2a\):
\[ (a^2 + 2a + 15)(a^2 + 2a - 3) \]

### Final Expression
Thus, the expression \((a^2 + 2a)^2 + 12(a^2 + 2a) - 45\) can be expressed as:
\[ (a^2 + 2a + 15)(a^2 + 2a - 3) \]

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