a) \(\Delta A B C\) vuông tại \(A\) suy ra \(\hat{B A C} = 9 0^{\circ}\) suy ra \(\hat{D A E} = 9 0^{\circ}\).

Do \(H D ⊥ A B\) suy ra \(\hat{H D A} = 9 0^{\circ}\); \(H E ⊥ A C\) suy ra \(\hat{H E A} = 9 0^{\circ}\).

Tứ giác \(A D H E\) có \(\hat{D A E} = \hat{H D A} = \hat{H E A} = 9 0^{\circ}\) suy ra tứ giác \(A D H E\) là hình chữ nhật. 

b) Do \(\Delta A H D\) vuông tại \(D\), áp dụng định lí Pythagore suy ra:

\(A H^{2} = A D^{2} + D H^{2}\)

\(25 = 16 + D H^{2}\)

\(D H^{2} = 9\) nên \(D H = 3\) cm.

Do \(A D H E\) là hình chữ nhật suy ra \(S_{A D H E} = A D . D H = 4.3 = 12\) (cm\(^{2}\)).

Vì đồ thị hàm số \(y = a x + b\) đi qua điểm \(A \left(\right. - 1 ; 2 \left.\right)\) nên ta có:

   \(2 = - 1. a + b\) suy ra \(- a + b = 2\)

Vi đồ thị hàm số \(y = a x + b\) đi qua điểm \(B \left(\right. 1 ; 4 \left.\right)\) nên ta có:

   \(4 = 1. a + b\) suy ra \(a + b = 4 \left(\right. 2 \left.\right)\)

Từ (1) và (2) ta tìm được \(a = 1 ; b = 3\)

Vậy hàm số cần tìm là \(y = x + 3\).

a) \(5 \left(\right. x + 2 y \left.\right) - 15 x \left(\right. x + 2 y \left.\right) = 5 \left(\right. x + 2 y \left.\right) . \left(\right. 1 - 3 x \left.\right)\).

b) \(4 x^{2} - 12 x + 9 = \left[\right. \left(\left(\right. 2 x \left.\right)\right)^{2} - 2.2 x . 3 + 3^{2} \left]\right. = \left(\left(\right. 2 x - 3 \left.\right)\right)^{2}\).

c) \(\left(\left(\right. 3 x - 2 \left.\right)\right)^{3} - 3 \left(\right. x - 4 \left.\right) \left(\right. x + 4 \left.\right) + \left(\left(\right. x - 3 \left.\right)\right)^{3} - \left(\right. x + 1 \left.\right) \left(\right. x^{2} - x + 1 \left.\right)\)

\(= 27 x^{3} - 54 x^{2} + 36 x - 8 - 3 \left(\right. x^{2} - 16 \left.\right) + x^{3} - 9 x^{2} + 27 x - 27 - \left(\right. x^{3} + 1 \left.\right)\)

\(= \left(\right. 27 x^{3} + x^{3} - x^{3} \left.\right) + \left(\right. - 54 x^{2} - 3 x^{2} - 9 x^{2} \left.\right) + \left(\right. 36 x + 27 x \left.\right) + \left(\right. - 8 + 48 - 27 - 1 \left.\right)\)

\(= 27 x^{3} - 66 x^{2} + 63 x + 12\).

My favorite hobby is playing the guitar. I started it three years ago when I got my first guitar as a birthday gift. Most of the time, I practice by myself, but sometimes I play with my best friend who also loves music. I usually practice at home in the evening after finishing my homework, or at school when we have music club activities. I think playing the guitar is really enjoyable because it helps me relax, express my feelings, and share music with others. It also makes my free time meaningful.