<s>[INST] How to factorize $2x^2+5x+3$? I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$
By trial, using ac-method I got stuck:
$$
ac = (2)(3) = 6\\
6 + ? = 5 \Rightarrow~ ? = 5 - 6 = -1
$$
Then,
$$2x^2+6x-x+3 = 2x(x+3)-x+3$$
At this point I could not get the answer, any help?
 [/INST] In general
$$ax^2+bx+c=a(x-x_1)(x-x_2)$$where $x_1,x_2$ are the roots of equation
$$ax^2+bx+c=0$$
in case $a=2,b=5,c=3$ and roots are
$$x_{1,2}=\frac{-5\pm\sqrt{5^2-4\cdot2\cdot3}}{2\cdot2}=\frac{-5\pm1}{4},x_1=-1,x_2=-\frac{3}{2}$$ and put these above we get
$$2x^2+5x+3=2(x-(-1))(x-(-\frac{3}{2}))=$$
$$=2(x+1)(x+\frac{3}{2})=(x+1)(2x+2\cdot\frac{3}{2})=(x+1)(2x+3)$$
</s>