proof verification - Mathematical Induction for sum of squares

I don't understand what I am doing wrong. I worked through this problem before and got it right, but then I misplaced a portion of the answer...I am having difficulties proving the second step in the induction process, substituting n for $k+1$.

This is the complete answer above, and I can get up to here the following

$$\frac{(k + 1)2k^2 + 7k + 6}{6}$$

However when I do the quadratic formula, I get

$$\frac{(k+1)(k-2)(k-1.5)}{6} \ne \frac{(k+1)(k+2)(2k+3)}{6}$$

What am I doing wrong with my factoring?

Answer

Solutions to equation $2k^2+7k+6=0$ are $k_1=-2$ and $k_2=-3/2$. Now

$$2k^2+7k+6 = 2(k-k_1)(k-k_2)$$ so

$$2k^2+7k+6 = 2(k+2)(k+{3\over 2})=(k+2)(2k+3)$$