Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
$$-x^{2}-2x+35$$
Factor the expression by grouping. First, the expression needs to be rewritten as $-x^{2}+ax+bx+35$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-2$$ $$ab=-35=-35$$
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 $-35$.
$$1,-35$$ $$5,-7$$
Calculate the sum for each pair.
$$1-35=-34$$ $$5-7=-2$$
The solution is the pair that gives sum $-2$.
$$a=5$$ $$b=-7$$
Rewrite $-x^{2}-2x+35$ as $\left(-x^{2}+5x\right)+\left(-7x+35\right)$.
$$\left(-x^{2}+5x\right)+\left(-7x+35\right)$$
Factor out $x$ in the first and $7$ in the second group.
$$x\left(-x+5\right)+7\left(-x+5\right)$$
Factor out common term $-x+5$ by using distributive property.
$$\left(-x+5\right)\left(x+7\right)$$
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 $ax^{2}+bx+c=0$.
$$-x^{2}-2x+35=0$$
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.
