42x² - 71x + 30

a = 42, b = - 71, c = 30

Find two numbers whose product is 42 and another two numbers whose product is 30.
      Factors of 42          Factors of 30
     1, 42 and 6, 7          - 5, - 6

Use FOIL method to check the middle term in the trial factors.
    Trial factors           Middle term
    (x - 5)(42x - 6)     - 6x - 210x = - 216x
    (6x - 5)(7x - 6)    - 36x - 35x = - 71x

So, 42x² - 71x + 30 = (6x - 5)(7x - 6).

3y² - 10y - 8

a = 3, b = - 10, c = - 8

Find two numbers whose product is 3 and another two numbers whose product is - 8.
    Factors of 3       Factors of - 8
         1, 3           - 1, 8; - 2, 4 and - 4, 2

Use FOIL method to check the middle term in the trial factors.
    Trial factors           Middle term
    (y - 1)(3y + 8)      8y - 3y = 5y
    (y - 2)(3y + 4)     4y - 6y = - 2y
    (y - 4)(3y + 2)     2y - 12y = - 10y

So, 3y² - 10y - 8 = (y - 4)(3y + 2).

3x² - 10xy - 25y²

a = 3, b = - 10, c = - 25

Find two numbers whose product is 3 and another two numbers whose product is - 25.
    Factors of 3        Factors of - 25
         1, 3            1, - 25 and 5, - 5

Use FOIL method to check the middle term in the trial factors.
    Trial factors           Middle term
    (x + y)(3x - 25y)      - 25xy + 3xy = - 22xy
    (x + 5y)(3x - 5y)      - 5xy + 15xy = 10xy
    (3x + 5y)(x - 5y)     - 15xy + 5xy = - 10xy

So, 3x² - 10xy - 25y² = (3x + 5y)(x - 5y).

x2 + 3x - 10 is in the form of x2 + bx + c. So, b = 3 and c = - 10.

x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

The required values of p and q are - 2, 5.

Therefore, x2 + 3x - 10 = (x - 2)(x + 5)

x2 + 13x + 40 is in the form of x2 + bx + c. So, b = 13 and c = 40.

x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

The required values of p and q are 8, 5.

Therefore, x2 + 13x + 40 = (x + 8)(x + 5).

y2 - 3y - 70 is in the form of x2 + bx + c. So, b = - 3 and c = - 70.

x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

The required values of p and q are - 10, 7.

Therefore, y2 - 3y - 70 = (y - 10)(y + 7).

Let the width of the ground = x ft.

Length of the ground = (x + 6) ft.

Area of the ground = 55 sq.ft.

So, x(x + 6) = 55

x2 + 6x = 55

x2 + 6x - 55 = 0

The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.

Compare the left side of the equation with x2 + bx + c to get b and c values. So, b = 6 and c = - 55.

Find the numbers p and q whose product is - 55 and whose sum is 6.

The required values of p and q are 11, - 5.

x2 + 6x - 55 = (x + 11)(x - 5)

So, the equation x2 + 6x - 55 = 0 can be written as (x + 11)(x - 5) = 0

x + 11 = 0 or x - 5 = 0

x = - 11 or x = 5

The width of the ground = 5 ft.

The length of the ground = 5 + 6 = 11 ft.

The factors of a trinomial x² + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.

Compare the equation with x² + bx + c to get b and c values. So, b = 6 and c = 9.

Find the numbers p and q whose product is 9 and sum is 6.

p and q    p + q
            1, 9        10
            3, 3          6

The required values of p and q are 3 and 3.

So, x² + 6x + 9 = (x + 3)(x + 3).

x² - 4x + 3 is in the form of x² + bx + c. So, b = - 4 and c = 3.

x² + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

	p and q	p + q	p × q
	- 1, - 3	- 4	3

The required values of p and q are - 1 and - 3.

So, x² - 4x + 3 = (x - 1)(x - 3).

The factors of a trinomial x² + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.

Compare the equation with x² + bx + c to get b and c values. So, b = 2 and c = -15.

Since c is negative, find the numbers p and q with different signs, whose product is -15 and sum is 2.

p and q    p + q
            -1, 15         14
            -3, 5           2

The required values of p and q are -3 and 5.

So, x² + 2x - 15 = (x - 3)(x + 5).

x² - 4x - 21 is in the form of x² + bx + c. So, b = - 4 and c = - 21.

x² + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

The required values of p and q are - 7 and 3.

So, x² - 4x - 21 =(x - 7) (x + 3).

The factors of a trinomial x² + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.

Compare the equation with x² + bx + c to get b and c values. So, b = -3 and c = -28.

Since c is negative, find the numbers p and q with different signs, whose product is -28 and sum is -3.

p and q    p + q
            - 4, 7           3
            - 7, 4         - 3

The required values of p and q are -7 and 4.

So, x² - 3x - 28 = (x - 7)(x + 4).

The factors of a trinomial x² + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.

Compare the equation with x² + bx + c to get b and c values. So, b = -7 and c = 10.

Since c is positive, find the numbers p and q with the same sign, whose product is 10 and sum is -7.

p and q    p + q
-5, -2           -7

The required values of p and q are -5 and -2.

So, x² - 7x + 10 = (x - 5)(x - 2).

x2 + 5x + 6 is in the form of x2 + bx + c. So, b = 5, c = 6.

x2 + bx + c can be factored as (x + p) (x + q), where b = p + q and c = pq.

p and q      p + q      p × q

 1, 6           7           6

 2, 3              5            6

The required values of p and q are 2 and 3.

Therefore, x2 + 5x + 6 = (x + 2)(x + 3).

x2 - 11x + 24 is in the form of x2 + bx + c. So, b = - 11 and c = 24.

x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.

The required values of p and q are - 3, - 8.

Therefore, x2 - 11x + 24 = (x - 3)(x - 8).