Solve (x^2-22x+121)-4x^2

$$( x ^ { 2 } - 22 x + 121 ) - 4 x ^ { 2 }$$

Factor

$$-\left(3x-11\right)\left(x+11\right)$$

Solution Steps

Multiply and combine like terms.

$$-3x^{2}-22x+121$$

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx+121. To find a and b, set up a system to be solved.

$$a+b=-22 ab=-3\times 121=-363$$

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 -363.

$$1,-363 3,-121 11,-33$$

Calculate the sum for each pair.

$$1-363=-362 3-121=-118 11-33=-22$$

The solution is the pair that gives sum -22.

$$a=11 b=-33$$

Rewrite -3x^{2}-22x+121 as \left(-3x^{2}+11x\right)+\left(-33x+121\right).

$$\left(-3x^{2}+11x\right)+\left(-33x+121\right)$$

Factor out -x in the first and -11 in the second group.

$$-x\left(3x-11\right)-11\left(3x-11\right)$$

Factor out common term 3x-11 by using distributive property.

$$\left(3x-11\right)\left(-x-11\right)$$

Evaluate

$$-\left(3x-11\right)\left(x+11\right)$$

Solution Steps

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-22x+121$$