calculus - Quadratic Formula Returns Different Root Signs

My question is as follows:

I am currently working on a problem set for "Integration of Rational Functions By Partial Fractions" and I came across the following problem:

$$\int_0^1 \frac{2}{2x^2+3x+1} \,dx$$

Now, the issue I have is with factoring the demoninator.

When I used the quadratic formula:

$$\begin{equation*} x = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a} \end{equation*}$$

I came up with the roots:

$$\biggl(x-\frac{1}{2}\biggl)\biggl(x-1\biggl)$$

I then multiplied the left hand root by 2:

$$(2x-1)(x-1)$$

However, when I multiplied out these two roots, I came out with:

$$2x^2-3x+1$$

I know I am most likely doing something wrong, but I have looked for other questions similar to this one and I have been having a hard time finding out an accurate explanation of what my error is in this problem.

(edit)

The format for quadratic roots is as follows:

$$(x-a)(x-b)$$

Therefore, if either a or b is negative, then the resulting root would be positive.

As a result, the two roots would therefore be:

$$(2x+1)(x+1)$$

Thanks

Answer

HINT: it is

$$2x^2+3x+1=(x+1)(2x+1)$$