The answer is letter B but I got letter D for my answer... What did I do wrong?
Chiro's suggestion for looking for common factors is a good one, but your method is fine as well.
When you get to $\displaystyle \displaystyle \begin{align*} 10y^2 - 13y - 3 \end{align*}$, you should note that $\displaystyle \displaystyle \begin{align*} 10(-3) = -30 \end{align*}$, so the two numbers that multiply to give -30 and add to give -13 are -15 and 2. So you should get
$\displaystyle \displaystyle \begin{align*} 10y^2 - 13y - 3 &= 10y^2 + 2y - 15y - 3 \\ &= 2y \left( 5y + 1 \right) - 3 \left( 5y + 1 \right) \\ &= \left( 5y + 1 \right) \left( 2y - 3 \right) \end{align*}$