Questions: Factor the polynomial completely. If the polynomial is not factorable, write "prime" as the answer. 2x^2+9x+4

Transcript text: Factor the polynomial completely. If the polynomial is not factorable, write "prime" as the answer. \[ 2 x^{2}+9 x+4 \]

Solution

Solution Steps

Step 1: Identify the polynomial

The given polynomial is: \[ 2x^{2} + 9x + 4 \]

Step 2: Find two numbers that multiply to \(2 \times 4 = 8\) and add to \(9\)

We need to find two numbers that multiply to \(8\) and add to \(9\). The numbers \(8\) and \(1\) satisfy this condition because: \[ 8 \times 1 = 8 \quad \text{and} \quad 8 + 1 = 9 \]

Step 3: Rewrite the middle term using the two numbers

Rewrite the polynomial by splitting the middle term: \[ 2x^{2} + 8x + x + 4 \]

Step 4: Factor by grouping

Group the terms and factor out the common factors: \[ (2x^{2} + 8x) + (x + 4) = 2x(x + 4) + 1(x + 4) \]

Step 5: Factor out the common binomial

Factor out the common binomial \((x + 4)\): \[ (2x + 1)(x + 4) \]

Final Answer

The polynomial is factored completely as: \[ \boxed{(2x + 1)(x + 4)} \]

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