Factor the expression by grouping. First, the expression needs to be rewritten as $9x^{2}+ax+bx+30$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-37$$ $$ab=9\times 30=270$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $270$.
Rewrite $9x^{2}-37x+30$ as $\left(9x^{2}-27x\right)+\left(-10x+30\right)$.
$$\left(9x^{2}-27x\right)+\left(-10x+30\right)$$
Factor out $9x$ in the first and $-10$ in the second group.
$$9x\left(x-3\right)-10\left(x-3\right)$$
Factor out common term $x-3$ by using distributive property.
$$\left(x-3\right)\left(9x-10\right)$$
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 $ax^{2}+bx+c=0$.
$$9x^{2}-37x+30=0$$
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 $x=\frac{37±17}{18}$ when $±$ is plus. Add $37$ to $17$.
$$x=\frac{54}{18}$$
Divide $54$ by $18$.
$$x=3$$
Now solve the equation $x=\frac{37±17}{18}$ when $±$ is minus. Subtract $17$ from $37$.
$$x=\frac{20}{18}$$
Reduce the fraction $\frac{20}{18}$ to lowest terms by extracting and canceling out $2$.
$$x=\frac{10}{9}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $3$ for $x_{1}$ and $\frac{10}{9}$ for $x_{2}$.
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 $9$
$$x ^ 2 -\frac{37}{9}x +\frac{10}{3} = 0$$
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 $rs = C$
$$r + s = \frac{37}{9} $$ $$ rs = \frac{10}{3}$$
Two numbers $r$ and $s$ sum up to $\frac{37}{9}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{37}{9} = \frac{37}{18}$. 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 $u$.
