Solve -x^2-4x+96-51

$$- { x }^{ 2 } -4x+96-51$$

Factor

$$\left(5-x\right)\left(x+9\right)$$

Solution Steps

Multiply and combine like terms.

$$-x^{2}-4x+45$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+45. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-45=-45$$

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.

$$a=5 b=-9$$

Rewrite -x^{2}-4x+45 as \left(-x^{2}+5x\right)+\left(-9x+45\right).

$$\left(-x^{2}+5x\right)+\left(-9x+45\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x+5\right)+9\left(-x+5\right)$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(x+9\right)$$

Evaluate

$$\left(5-x\right)\left(x+9\right)$$

Solution Steps

Subtract 51 from 96 to get 45.

$$-x^{2}-4x+45$$