Factor the expression by grouping. First, the expression needs to be rewritten as $5x^{2}+ax+bx-9$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-4$$ $$ab=5\left(-9\right)=-45$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-45$.
$$1,-45$$ $$3,-15$$ $$5,-9$$
Calculate the sum for each pair.
$$1-45=-44$$ $$3-15=-12$$ $$5-9=-4$$
The solution is the pair that gives sum $-4$.
$$a=-9$$ $$b=5$$
Rewrite $5x^{2}-4x-9$ as $\left(5x^{2}-9x\right)+\left(5x-9\right)$.
$$\left(5x^{2}-9x\right)+\left(5x-9\right)$$
Factor out $x$ in $5x^{2}-9x$.
$$x\left(5x-9\right)+5x-9$$
Factor out common term $5x-9$ by using distributive property.
$$\left(5x-9\right)\left(x+1\right)$$
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 $ax^{2}+bx+c=0$.
$$5x^{2}-4x-9=0$$
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 $x=\frac{4±14}{10}$ when $±$ is plus. Add $4$ to $14$.
$$x=\frac{18}{10}$$
Reduce the fraction $\frac{18}{10}$ to lowest terms by extracting and canceling out $2$.
$$x=\frac{9}{5}$$
Now solve the equation $x=\frac{4±14}{10}$ when $±$ is minus. Subtract $14$ from $4$.
$$x=-\frac{10}{10}$$
Divide $-10$ by $10$.
$$x=-1$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{9}{5}$ for $x_{1}$ and $-1$ for $x_{2}$.
Cancel out $5$, the greatest common factor in $5$ and $5$.
$$5x^{2}-4x-9=\left(5x-9\right)\left(x+1\right)$$
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 $5$
$$x ^ 2 -\frac{4}{5}x -\frac{9}{5} = 0$$
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 $rs = C$
$$r + s = \frac{4}{5} $$ $$ rs = -\frac{9}{5}$$
Two numbers $r$ and $s$ sum up to $\frac{4}{5}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{4}{5} = \frac{2}{5}$. 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 $u$.
$$r = \frac{2}{5} - u$$ $$s = \frac{2}{5} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -\frac{9}{5}$
