Factor the expression by grouping. First, the expression needs to be rewritten as $4x^{2}+ax+bx-21$. To find $a$ and $b$, set up a system to be solved.
$$a+b=5$$ $$ab=4\left(-21\right)=-84$$
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 $-84$.
Rewrite $4x^{2}+5x-21$ as $\left(4x^{2}-7x\right)+\left(12x-21\right)$.
$$\left(4x^{2}-7x\right)+\left(12x-21\right)$$
Factor out $x$ in the first and $3$ in the second group.
$$x\left(4x-7\right)+3\left(4x-7\right)$$
Factor out common term $4x-7$ by using distributive property.
$$\left(4x-7\right)\left(x+3\right)$$
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 $ax^{2}+bx+c=0$.
$$4x^{2}+5x-21=0$$
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 $x=\frac{-5±19}{8}$ when $±$ is plus. Add $-5$ to $19$.
$$x=\frac{14}{8}$$
Reduce the fraction $\frac{14}{8}$ to lowest terms by extracting and canceling out $2$.
$$x=\frac{7}{4}$$
Now solve the equation $x=\frac{-5±19}{8}$ when $±$ is minus. Subtract $19$ from $-5$.
$$x=-\frac{24}{8}$$
Divide $-24$ by $8$.
$$x=-3$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{7}{4}$ for $x_{1}$ and $-3$ 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 $4$
$$x ^ 2 +\frac{5}{4}x -\frac{21}{4} = 0$$
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 $rs = C$
$$r + s = -\frac{5}{4} $$ $$ rs = -\frac{21}{4}$$
Two numbers $r$ and $s$ sum up to $-\frac{5}{4}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{5}{4} = -\frac{5}{8}$. 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 $u$.
$$r = -\frac{5}{8} - u$$ $$s = -\frac{5}{8} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -\frac{21}{4}$
