If \(2 x^{2}+5 x+2=(x+2) \times\) a polynomial what must be the coefficient of \(x\) in this unknown polynomial?

First, we need to set up the polynomial long division. Write \((x+2)\) as the divisor, and \(2x^2 + 5x + 2\) as the dividend: ________________ (x + 2) |(2x^2 + 5x + 2)

In this step, we will perform the polynomial long division as follows: 1. Divide the term with the highest power in the dividend (\(2x^2\)) by the term with the highest power in the divisor (\(x\)), which results in \(2x\). 2. Write \(2x\) in the quotient, and multiply every term of the divisor by \(2x\), and subtract the resulting expression from the dividend. 2x_____________ (x + 2) |(2x^2 + 5x + 2) - (2x^2 + 4x) __________________ x + 2 3. Repeat the process for the remaining terms in the dividend. Divide the term with the highest power (\(x\)) by the term with the highest power in the divisor (\(x\)), which results in \(1\). 4. Write \(1\) in the quotient, multiply every term of the divisor by \(1\), and subtract the resulting expression from the previous remainder. 2x + 1____________ (x + 2) |(2x^2 + 5x + 2) - (2x^2 + 4x) __________________ x + 2 - (x + 2) __________________ 0 From the long division, we can see that the quotient is \(2x+1\), which means that \((x+2)\) times \((2x+1)\) equals to the given polynomial \(2x^2+5x+2\).

Now that we have found the unknown polynomial by performing the polynomial long division, we can directly see the coefficient of the \(x\) term in the quotient. The polynomial is \(2x + 1\), and the coefficient of the \(x\) term is \(\boxed{2}\).