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

\(\Leftrightarrow x+2=-\left(8x^3+12x^2+6x+1\right)\)

\(\Leftrightarrow8x^3+12x^2+7x+3=0\)

\(\Leftrightarrow\left(8x^3+8x^2\right)+\left(4x^2+4x\right)+\left(3x+3\right)=0\)

\(\Leftrightarrow8x^2\left(x+1\right)+4x\left(x+1\right)+3\left(x+1\right)=0\)

\(\Leftrightarrow\left(8x^2+4x+3\right)\left(x+1\right)=0\)

\(\Leftrightarrow x+1=0\) (vì \(8x^2+4x+3=8\left(x^2+\dfrac{1}{2}x+\dfrac{3}{8}\right)=8\left[\left(x^2+2.\dfrac{1}{4}x+\dfrac{1}{16}\right)+\dfrac{5}{16}\right]=8\left(x+\dfrac{1}{4}\right)^2+\dfrac{5}{2}>0\))

\(\Leftrightarrow x=-1\)

Vậy \(x=-1\)

I.
1. a. centre
2. d. charity
3. b. volunteer
4. c. general
5. a. provide
II.
1. d. certain
2. c. gym
3. a. colour
4. b. geography
5. a. charitable
III.
1. a. about
2. b. to
3. b. collected
4. a. clean it up
5. b. Homeless
6. c. already
7. c. need
8. a. because
9. a. took
10. a. air pollution

Last year I was a junior in high school. At present,I am a senior in high school

\(C=x^7-26x^6+27x^5-47x^4-77x^3+50x^2+x-24\)

\(=x^7-\left(25+1\right)x^6+\left(25+2\right)x^5-\left(2.25-3\right)x^4-\left(3.25+2\right)x^3+2.25x^2+x-24\)

\(=x^7-\left(x+1\right)x^6+\left(x+2\right)x^5-\left(2x-3\right)x^4-\left(3x+2\right)x^3+2x^3+25-24\) (vì \(x=25\))

\(=x^7-x^7-x^6+x^6+2x^5-2x^5+3x^4-3x^4-2x^3+2x^3+1\)

\(=1\)

6.He had his bike repaired by his father yesterday.
 He got his father to repair his bike yesterday.
7.Nga’s mother makes her study hard.
 Nga is made to study hard by her mother.
8.We need to repair the house.
 The house needs to be repaired by us.
9. She works very hard, so she is very successful.
 The harder she works, the more successful she is.
1O. My English friend finds using chopsticks difficult.
=> My English friend isn't used to using chopsticks.
1. It took us 3 hours to open the door.
=> We spent 3 hours opening the door.
2. We haven’t visited our grandparents for two months.
=> The last time we visited our grandparents was two months ago.
3. “Would you mind getting out of the car?” the driver said to me.
=> The driver asked me to get out of the car.
4. She is both intelligent and beautiful.
=> She is not only intelligent but also beautiful.
5. People think the jewels were stolen by one of the guests.
=> One of the guests is thought to have stolen the jewels.
6. This car costs $10,000 and so does that one.
=> This car is as expensive as that one.
8. I haven’t seen that man here before.
=> This is the first time I have seen that man here.
9. Please don’t make any noise . I’m very tired
=> I’d rather you didn’t make any noise because I'm tired.
10. He did not drive because of the thick fog.
=> The thick fog prevented him from driving.

1. He did the homework in two hours.
It took him two hours to do the homework.
2. I am sorry I don’t have time to come and help you with your homework.
I wish I had time to come and help you with your homework.
3. The school keeper opens the gate every day.
The gate is opened by the school keeper every day.
4. “Why don’t we go out for a while?”
He suggested going out for a while.
5. He forgot his umbrella, so he got wet.
He got wet because he forgot his umbrella.
6. Disabled people find traveling by bus difficult.
It is difficult for disabled people to travel by bus.
7. Please don’t ask me that question.
I’d rather you didn't ask me that question.
8. The last time I drank beer was five months ago.
I haven't drunk beer for five months.
9. My car needs to be washed.
I need to have my car washed.
10. Although the questions were difficult, Lan answered them perfectly
Despite the difficult questions, Lan answered them perfectly.

\(x^2+10xy+25y^2\)

\(=x^2+2.x.5y+\left(5y\right)^2\)

\(=\left(x+5y\right)^2\)

1. You shouldn't have been driving so carelessly! You (kill) could have killed someone.

2. (Not eat) Not having eaten that dish before, I found it quite strange.

3. When they came back from the beach, their skin was red. They (lie) had lied in the sun for 5 hours.

4. They discussed the project, then went on (discuss) discussing another topic.

5. By the time the train arrives, we (wait) will have waited here for more than an hour.

6. If you want to catch the last train, it’s time you (leave) left.

7. It is important that he (understand) understand it.

c) Ta có:

\(VT=\left(a+b+c\right)^2+\left(a-b+c\right)^2+\left(a+b-c\right)^2+\left(-a+b+c\right)^2\)

\(=\left(a^2+b^2+c^2+2ab+2bc+2ca\right)+\left(a^2+b^2+c^2-2ab-2bc+2ca\right)+\left(a^2+b^2+c^2+2ab-2bc-2ca\right)+\left(a^2+b^2+c^2-2ab+2bc-2ca\right)\)

\(=a^2+b^2+c^2+2ab+2bc+2ca+a^2+b^2+c^2-2ab-2bc+2ca+a^2+b^2+c^2+2ab-2bc-2ca+a^2+b^2+c^2-2ab+2bc-2ca\)

\(=4a^2+4b^2+4c^2\)

\(=4\left(a^2+b^2+c^2\right)=VP\) (đpcm)

d) Ta có:

\(VT=\left(x+y\right)^4+x^4+y^4\)

\(=\left(x^4+4x^3y+6x^2y^2+4xy^3+y^4\right)+x^4+y^4\)

\(=x^4+4x^3y+6x^2y^2+4xy^3+y^4+x^4+y^4\)

\(=2x^4+4x^3y+6x^2y^2+4xy^3+2y^4\)

\(VP=2\left(x^2+xy+y^2\right)^2\)

\(=2\left(x^4+x^2y^2+y^4+2x^3y+2xy^3+2x^2y^2\right)\)

\(=2x^4+2x^2y^2+2y^4+4x^3y+4xy^3+4x^2y^2\)

\(=2x^4+4x^3y+6x^2y^2+4xy^3+2y^4\)

Từ đó, suy ra: \(VT=VP\) (đpcm)

e) Ta có:

\(VT=\left(5a-3b+8c\right)\left(5a-3b-8c\right)\)

\(=\left[\left(5a-3b\right)+8c\right]\left[\left(5a-3b\right)-8c\right]\)

\(=\left(5a-3b\right)^2-64c^2\)

\(=\left(5a-3b\right)^2-16.4c^2\)

\(=\left(5a-3b\right)^2-16\left(a^2-b^2\right)\) (vì \(a^2-b^2=4c^2\))

\(=25a^2-30ab+9b^2-16a^2+16b^2\)

\(=9a^2-30ab+25b^2\)

\(=\left(3a-5b\right)^2=VP\) (đpcm)

f) Theo giả thiết, ta có:

\(\left(a+b+c\right)^2=3\left(ab+bc+ca\right)\)

\(a^2+b^2+c^2+2ab+2bc+2ca=3ab+3bc+3ca\)

\(a^2+b^2+c^2=ab+bc+ca\)

\(2a^2+2b^2+2c^2=2ab+2bc+2ca\)

\(\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)

\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\Rightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a-b=0\\b-c=0\\c-a=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a=b\\b=c\\c=a\end{matrix}\right.\)

\(\Rightarrow a=b=c\) (đpcm)

1. The country is very mountainous so travelling by road is difficult. (MOUNTAIN)

2. The teacher stressed the need for regular attendance. (ATTEND)

3. Sarah cautiously opened the door of the cellar, wondering what she might find. (CAUTION)

4. There were over fifty musicians in the orchestra. (MUSIC)

5. Jim always does what he says; he's a very reliable person. (RELY)

6. Alexander knows which mushrooms are poisonous, so ask him before you pick them. (POISON)

7. The company is very efficient and gives a speedy service. (SPEED)

8. The warmness of the fire was very welcome after our long walk. (WARM)

9. They have agreed to widen the road because of the amount of heavy traffic now using it. (WIDE)

10. John's behaviour improved at his new school. (BEHAVE)