Factor the expression by grouping. First, the expression needs to be rewritten as $21x^{2}+ax+bx-8$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-22$$ $$ab=21\left(-8\right)=-168$$
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 $-168$.
Rewrite $21x^{2}-22x-8$ as $\left(21x^{2}-28x\right)+\left(6x-8\right)$.
$$\left(21x^{2}-28x\right)+\left(6x-8\right)$$
Factor out $7x$ in the first and $2$ in the second group.
$$7x\left(3x-4\right)+2\left(3x-4\right)$$
Factor out common term $3x-4$ by using distributive property.
$$\left(3x-4\right)\left(7x+2\right)$$
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 $ax^{2}+bx+c=0$.
$$21x^{2}-22x-8=0$$
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 $x=\frac{22±34}{42}$ when $±$ is plus. Add $22$ to $34$.
$$x=\frac{56}{42}$$
Reduce the fraction $\frac{56}{42}$ to lowest terms by extracting and canceling out $14$.
$$x=\frac{4}{3}$$
Now solve the equation $x=\frac{22±34}{42}$ when $±$ is minus. Subtract $34$ from $22$.
$$x=-\frac{12}{42}$$
Reduce the fraction $\frac{-12}{42}$ to lowest terms by extracting and canceling out $6$.
$$x=-\frac{2}{7}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{4}{3}$ for $x_{1}$ and $-\frac{2}{7}$ for $x_{2}$.
Multiply $\frac{3x-4}{3}$ times $\frac{7x+2}{7}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
