Question

Find all solutions to the equation (x² + 5x + 6)(x² − 3x − 4) = 0.

Answer 1

The solutions are:

`x=-2,\:x=-3,\:x=-1,\:x=4`

Explanation:

To find the solutions to the equation
`\left(x^2+5x+6\right)\left(x^2-3x-4\right)=0` you need to:

• Factor
  `\left(x^2+5x+6\right)`

Break the expression into groups

`x^2+5x+6=\left(x^2+2x\right)+\left(3x+6\right)`

Factor out
`x` from
`(x^2+2x)`

`x^2+2x=x\left(x+2\right)`

Factor out 3 from
`3x+6`

`3x+6=3\left(x+2\right)`

`x^2+5x+6=x\left(x+2\right)+3\left(x+2\right)\\\\\mathrm{Factor\:out\:common\:term\:}x+2\\\\x^2+5x+6=\left(x+2\right)\left(x+3\right)`

• Factor
  `\left(x^2-3x-4\right)`

`x^2-3x-4=\left(x^2+x\right)+\left(-4x-4\right)\\\\x^2-3x-4=x\left(x+1\right)-4\left(x+1\right)\\\\x^2-3x-4=\left(x+1\right)\left(x-4\right)`

Therefore

`\left(x^2+5x+6\right)\left(x^2-3x-4\right)=\left(x+2\right)\left(x+3\right)\left(x+1\right)\left(x-4\right)=0`

Using the Zero Factor Theorem: