Factor the expression by grouping. First, the expression needs to be rewritten as $15y^{2}+ay+by+4$. To find $a$ and $b$, set up a system to be solved.
$$a+b=16$$ $$ab=15\times 4=60$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. List all such integer pairs that give product $60$.
Rewrite $15y^{2}+16y+4$ as $\left(15y^{2}+6y\right)+\left(10y+4\right)$.
$$\left(15y^{2}+6y\right)+\left(10y+4\right)$$
Factor out $3y$ in the first and $2$ in the second group.
$$3y\left(5y+2\right)+2\left(5y+2\right)$$
Factor out common term $5y+2$ by using distributive property.
$$\left(5y+2\right)\left(3y+2\right)$$
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 $ax^{2}+bx+c=0$.
$$15y^{2}+16y+4=0$$
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 $y=\frac{-16±4}{30}$ when $±$ is plus. Add $-16$ to $4$.
$$y=-\frac{12}{30}$$
Reduce the fraction $\frac{-12}{30}$ to lowest terms by extracting and canceling out $6$.
$$y=-\frac{2}{5}$$
Now solve the equation $y=\frac{-16±4}{30}$ when $±$ is minus. Subtract $4$ from $-16$.
$$y=-\frac{20}{30}$$
Reduce the fraction $\frac{-20}{30}$ to lowest terms by extracting and canceling out $10$.
$$y=-\frac{2}{3}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-\frac{2}{5}$ for $x_{1}$ and $-\frac{2}{3}$ for $x_{2}$.
Multiply $\frac{5y+2}{5}$ times $\frac{3y+2}{3}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
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 $15$
$$x ^ 2 +\frac{16}{15}x +\frac{4}{15} = 0$$
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 $rs = C$
$$r + s = -\frac{16}{15} $$ $$ rs = \frac{4}{15}$$
Two numbers $r$ and $s$ sum up to $-\frac{16}{15}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{16}{15} = -\frac{8}{15}$. 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 $u$.
