Solve the equation two times 𝑥 plus one squared plus five times 𝑥 plus one equals zero.
When I look at this equation, I immediately notice that both of the addends are being multiplied by 𝑥 plus one. If we undistribute 𝑥 plus one, we pull it out. And then we’re left with two times 𝑥 plus one. This is really important; that 𝑥 plus one was being squared. So if you remove one of the 𝑥 plus one factors, there’s still one left, like this: two times 𝑥 plus one.
However, five times 𝑥 plus one is now plus five. The 𝑥 plus one factor has been removed. Our new equation says 𝑥 plus one times two times 𝑥 plus one plus five. We have two factors. We want to take each of these factors and set them equal to zero. 𝑥 plus one equal zero. And two times 𝑥 plus one plus five equals zero. 𝑥 plus one equals zero means we can subtract one from both sides and 𝑥 is equal to negative one.
For our next factor, we subtract five from both sides of the equation. Positive five minus five equals zero. Zero minus five equals negative five. We now have two times 𝑥 plus one equals negative five. We distribute our two to both 𝑥 and the one. Now we have two 𝑥 plus two equals negative five.
We need to get rid of our plus two. So we subtract two from both sides of the equation. Plus two minus two cancels out. Negative five minus two equals negative seven. Two 𝑥 equals negative seven. From there, we divide the left side by two and the right side by to two.
Two divided by two cancels out. And 𝑥 is equal to negative seven-halves. The two solutions for this equation is 𝑥 equal negative one and 𝑥 equals negative seven- halves.