a) x² + 4x - 45

= (x² - 5x) + (9x - 45)

= x(x - 5) + 9(x - 5)

= (x + 9)(x - 5)

b) x² - 12xy - 13y²

= (x² - 13xy) + (xy - 13y²)

= x(x - 13y) + y(x - 13y)

= (x + y)(x - 13y)

c) 16 - x² - 2xy - y²

= 16 - (x² + 2xy + y²)

= 4² - (x + y)²

= (4 - x - y)(4 + x + y)

d) x - 2y + x² - 4y²

= (x - 2y) + (x² - 4y²)

= (x - 2y) + (x - 2y)(x + 2y)

= (x - 2y)(x + 2y + 1)

c. Phương trình <=> (4x² + 4x + 1) + (9x² + y² + 1 + 6xy + 2y + 6x) = 22 <=> (2x + 1)² + (3x + y + 1)² = 22

Vì (3x + y + 1)² \(\ge0\Rightarrow\left(2x+1\right)^2\le22\Leftrightarrow-\sqrt{22}\le2x+1\le\sqrt{22}\Leftrightarrow-\sqrt{22}-1\le2x\le\sqrt{22}-1\Leftrightarrow\dfrac{-\sqrt{22}-1}{2}\le x\le\dfrac{\sqrt{22}-1}{2}\)

Mà x nguyên nên \(x\in\left\{-2;-1;0;1\right\}.\)

Với \(x=-2,\) ta có: \(\left(-3\right)^2+\left(y-5\right)^2=22\Leftrightarrow y^2-10y+12=0.\) Phương trình này không có nghiệm nguyên.

Với \(x=-1,\) ta có: \(\left(-1\right)^2+\left(y-2\right)^2=22\Leftrightarrow y^2-4y-17=0.\) Phương trình này không có nghiệm nguyên.

Với \(x=0,\) ta có: \(1^2+\left(y+1\right)^2=22\Leftrightarrow y^2+2y-20=0.\) Phương trình này không có nghiệm nguyên.

Với \(x=1,\) ta có: \(3^2+\left(y+4\right)^2=22\Leftrightarrow y^2+4y-9=0.\) Phương trình này không có nghiệm nguyên.

Vậy phương trình không có nghiệm nguyên.

d. Phương trình \(\Leftrightarrow\left(x^2+9y^2+1-6xy-6y+2x\right)+\left(4y^2-4y+1\right)=101\Leftrightarrow\left(3y-x-1\right)^2+\left(2y-1\right)^2=101\)

Vì \(\left(3y-x-1\right)^2\ge0\Rightarrow\left(2y-1\right)^2\le101\Leftrightarrow-\sqrt{101}\le2y-1\le\sqrt{101}\Leftrightarrow1-\sqrt{101}\le2y\le1+\sqrt{101}\Leftrightarrow\dfrac{1-\sqrt{101}}{2}\le y\le\dfrac{1+\sqrt{101}}{2}\)

Mà y nguyên nên \(y\in\left\{-4;-3;-2;-1;0;1;2;3;4;5\right\}.\)

Với \(y=-4,\) ta có: \(\left(-x-13\right)^2+\left(-9\right)^2=101\Leftrightarrow\left(x+13\right)^2=20.\) Phương trình này không có nghiệm nguyên.

Với \(y=-3,\) ta có: \(\left(-x-10\right)^2+\left(-7\right)^2=101\Leftrightarrow\left(x+10\right)^2=52.\) Phương trình này không có nghiệm nguyên.

Với \(y=-2,\) ta có: \(\left(-x-7\right)^2+\left(-5\right)^2=101\Leftrightarrow\left(x+7\right)^2=76.\) Phương trình này không có nghiệm nguyên.

Với \(y=-1,\) ta có: \(\left(-x-4\right)^2+\left(-3\right)^2=101\Leftrightarrow\left(x+4\right)^2=92.\) Phương trình này không có nghiệm nguyên.

Với \(y=0,\) ta có: \(\left(-x-1\right)^2+\left(-1\right)^2=101\Leftrightarrow\left(x+1\right)^2=100\Leftrightarrow x+1=10\)  hoặc x + 1 = -10 <=> x = 9 (nhận) hoặc x = -11 (nhận)

Với \(y=1,\) ta có: \(\left(-x-2\right)^2+1=101\Leftrightarrow\left(x+2\right)^2=100\) <=> x + 2 = 10 hoặc x + 2 = -10 <=> x = 8 (nhận) hoặc x = -12 (nhận)

Với \(y=2,\) ta có: \(\left(5-x\right)^2+9=101\Leftrightarrow\left(x-5\right)^2=92.\) Phương trình này không có nghiệm nguyên.

Với \(y=3,\) ta có: \(\left(8-x\right)^2+25=101\Leftrightarrow\left(x-8\right)^2=76.\) Phương trình này không có nghiệm nguyên.

Với \(y=4,\) ta có: \(\left(11-x\right)^2+49=101\Leftrightarrow\left(x-11\right)^2=52.\) Phương trình này không có nghiệm nguyên.

Với \(y=5,\) ta có: \(\left(14-x\right)^2+9^2=101\Leftrightarrow\left(x-14\right)^2=20.\) Phương trình này không có nghiệm nguyên.

Vậy nghiệm của phương trình là: (x;y) = (9;0), (-11;0), (8;1), (-12;1).

1. We couldn't go out to eat last night as it rained heavily.

2. Sybil suggest that Minh Thai should use energy - saving appliances.

3. I’m very happy that my sister has good result.

4. If the temperature keeps increasing, ice on the poles will melt.

7. Hilary bought this car because of its low price.

8. Cecily suggest taking a rafting trip this weekend.

9. You learn English very hard.

52. That is the man who used to work with me when I lived in New York.

53. They are talking about the girl who used to be a Miss-World.

54. I told you about the boy who is very good at math.

55. The men whom the police have caught stole my car.

56. Mr. Pike who can play tennis is over seventy.

57. The boy whose hair is black curly is Nam.

58. Although the wind was cold this morning, we went for a walk.

59. Although he didn't finish his lesson, he went to bed.

60. Despite her serious illness, she enjoyed life very much.

1. worked
2. does he
3. to going
4. went
5. talking
6. eating
7. used
8. pleased
9. pollute
10. has worked

71. Thu told that he wasn't not good at speaking English.
72. Because of the polluted water, we can't swim in this river.
73. Since I am sick, I don't want to go out now.
74. That is the teacher who used to teach me when I lived in the town.
76. If she prepares carefully, she will do the test well.
77. If they support actively, we can finish the research perfectly.
78. If you work hard, you can pass the examination.
79. If she uses energy-saving bulb, she can save money.
80. It is very important to keep the environment clean.
81. Although English is not easy, I like to study it.

61. Even though he is intelligent, he doesn't study well at school.
62. Although it was hot in the afternoon, we decided not to go for a swim.
63. In spite of not having enough money, he bought this bike.
64. Though he was tired, he didn't say a word.
65. Although it snowed heavily, we got home on time.
66. He seems a friendly person; however, I didn't like him.
67. Sue is looking forward to going to Spain on holiday.
68. The leader suggested everything should be checked once more.
69. Jim suggested I should take a day off.
70. I suggested he should stop smoking.

91. Vinh apologized to Lan for losing her book.
92. Van suggested that I put a better lock on the door.
93. If you don't water these plants, they will die.
Unless you water these plants, they will die.
94. If I were rich, I could buy that house.
95. If he didn't work in the evening, he would have time to play with his children.
96. If they didn't use a lot of wood pulp, many trees would not be cut down.
97. If she could come, he would not be disappointed.
98. If disasters didn't happen more often, we would not get more trouble.
99. If I were you, I could apply for this special job.
100. If there were air and water there, there would be life on other planets.
101. If the tickets were not so expensive, we would go to the concert.

\(\sqrt{a}-1< 0\Leftrightarrow\sqrt{a}< 1\Leftrightarrow0\le a< 1\)

Vậy \(0\le a< 1.\)

c) \(\left(2x-4\right)^{2016}+\left|x^2-4\right|=0\Rightarrow\left\{{}\begin{matrix}\left(2x-4\right)^{2016}=0\\\left|x^2-4\right|=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}2x-4=0\\x^2-4=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}2x=4\\x^2=4\end{matrix}\right.\Rightarrow x=2\)

f) \(\left(3x-1\right)^2=\dfrac{4^{-1}-3.\left(\dfrac{3}{2}\right)^2}{5-\left(\dfrac{1}{3}\right)^{-1}}:\left[\left(\dfrac{6}{13}-\dfrac{1}{2}\right).2\right]\)

\(\left(3x-1\right)^2=\dfrac{\dfrac{1}{4}-3.\dfrac{9}{4}}{5-3}:\left[2.\dfrac{6}{13}-2.\dfrac{1}{2}\right]\)

\(\left(3x-1\right)^2=\dfrac{\dfrac{1}{4}-\dfrac{27}{4}}{2}:\left[\dfrac{12}{13}-1\right]\)

\(\left(3x-1\right)^2=\dfrac{-\dfrac{26}{4}}{2}:\left[-\dfrac{1}{13}\right]\)

\(\left(3x-1\right)^2=-\dfrac{13}{4}.\left[-13\right]\)

\(\left(3x-1\right)^2=\dfrac{169}{4}\)

\(\Rightarrow3x-1=\dfrac{13}{2}\) hoặc \(3x-1=-\dfrac{13}{2}\)

\(3x=\dfrac{15}{2}\) hoặc \(3x=-\dfrac{11}{2}\)

\(x=\dfrac{5}{2}\) hoặc \(x=-\dfrac{11}{6}\)