Factor the expression by grouping. First, the expression needs to be rewritten as $2y^{2}+ay+by-3$. To find $a$ and $b$, set up a system to be solved.
$$a+b=5$$ $$ab=2\left(-3\right)=-6$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-6$.
$$-1,6$$ $$-2,3$$
Calculate the sum for each pair.
$$-1+6=5$$ $$-2+3=1$$
The solution is the pair that gives sum $5$.
$$a=-1$$ $$b=6$$
Rewrite $2y^{2}+5y-3$ as $\left(2y^{2}-y\right)+\left(6y-3\right)$.
$$\left(2y^{2}-y\right)+\left(6y-3\right)$$
Factor out $y$ in the first and $3$ in the second group.
$$y\left(2y-1\right)+3\left(2y-1\right)$$
Factor out common term $2y-1$ by using distributive property.
$$\left(2y-1\right)\left(y+3\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$2y^{2}+5y-3=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $y=\frac{-5±7}{4}$ when $±$ is plus. Add $-5$ to $7$.
$$y=\frac{2}{4}$$
Reduce the fraction $\frac{2}{4}$ to lowest terms by extracting and canceling out $2$.
$$y=\frac{1}{2}$$
Now solve the equation $y=\frac{-5±7}{4}$ when $±$ is minus. Subtract $7$ from $-5$.
$$y=-\frac{12}{4}$$
Divide $-12$ by $4$.
$$y=-3$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1}{2}$ for $x_{1}$ and $-3$ for $x_{2}$.
Cancel out $2$, the greatest common factor in $2$ and $2$.
$$2y^{2}+5y-3=\left(2y-1\right)\left(y+3\right)$$
Steps Using Direct Factoring Method
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form $x^2+Bx+C=0$.This is achieved by dividing both sides of the equation by $2$
$$x ^ 2 +\frac{5}{2}x -\frac{3}{2} = 0$$
Let $r$ and $s$ be the factors for the quadratic equation such that $x^2+Bx+C=(x−r)(x−s)$ where sum of factors $(r+s)=−B$ and the product of factors $rs = C$
$$r + s = -\frac{5}{2} $$ $$ rs = -\frac{3}{2}$$
Two numbers $r$ and $s$ sum up to $-\frac{5}{2}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{5}{2} = -\frac{5}{4}$. You can also see that the midpoint of $r$ and $s$ corresponds to the axis of symmetry of the parabola represented by the quadratic equation $y=x^2+Bx+C$. The values of $r$ and $s$ are equidistant from the center by an unknown quantity $u$. Express $r$ and $s$ with respect to variable $u$.
$$r = -\frac{5}{4} - u$$ $$s = -\frac{5}{4} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -\frac{3}{2}$
