The solutions to the above-mentioned questions are:

**Solution 1**

In the question

Factors of x² = x * x

Factors of 6x = 2 * 3 * x

Factors of -16 = -2 * 8

So according to the question,

We know that 6x can be written as +-2x + 8x as the following expression will be equal to 6x.

x² + 6x - 16 = x² - 2x + 8x - 16

x( x – 2 ) + 8( x – 2 )

(x + 8) (x – 2)

Hence the factors of x2 + 6x – 16 = (x + 8) (x – 2)

**Solution 2**

i) (a + 6) × 2 = a² + 12a + 36

In this equation, the left-hand side (LHS) is

(a + 6)²

Applying the formula of (a + b)²,

(a + 6)² can be written as a² + 12a + 36

We know that the right-hand side (RHS) = a2 + 12a + 36, which is equal to the LHS (a² + 12a + 36)

Hence it is proved that LHS = RHS.

ii) (2a)² + 5a = 4a + 5a

Here, in this equation, we see that

LHS = (2a)² + 5a, which can be further written as 4a² + 5a on opening the brackets of (2a)²

We see that RHS = 4a + 5a

Both sides are not equal. Therefore, LHS ≠ RHS.

Thus the correct equation would be

(2a)² + 5a = 4a2 + 5a

**Solution 3**

i) The factors for 6 xyz are 2 * 3 * x * y * z

The factors for 24 xy² are 2 * 2 * 2 * 3 * x * y * y

The factors for 12 x²y are 2 * 2 * 3 * x * x * y

In the above simplified factors, we see that the common factors for 6 xyz, 24 xy², and 12 x²y are 2, 3, x, and y:

(2 * 3 * x * y) = 6xy

Common factors = 2, 3, x, y, or 6xy

ii) The factors for 3x² y³ are 3 * x * x * y * y * y

The factors for 10x³ y² are 2 * 5 * x * x * x * y * y

The factors for 6 x² y² z are 3 * 2 * x * x * y * y * z

In the above simplified factors, we see that the common factors for 3x² y³, 10x³ y², and 6x² y² z are x2, y2, and (x2 * y2) = x2 y2

Common factors = x², y², or x² y²

**Solution 4**

To solve the given expression, it is important to first expand (x + y)^{2}

To solve the expression, use this formula:

(x + y)² = x² + 2xy + y²

(x + y)² – 4xy can be written as

x² + 2xy + y² - 4xy, where we have substituted (x + y)² for x² + 2xy + y².

Thus the following result becomes

x² + y² – 2xy

We also know that the formula for

(x - y)² is x² + y² - 2xy

Hence we can substitute the result,

x² + y² – 2xy with (x + y)² - 4xy.

Thus, (x + y)² - 4xy becomes (x - y)²

**Solution 5**

i) 7x – 42

Taking 7 common in the equation

Now the factor is 7(x – 6)

ii) 6p – 12q

Taking 6 common in the equation

Now the factor is 6(p – 2q)

iii) 7a² + 14a

Taking 7a common in the equation

Now the factor is 7a(a + 2)

iv) -16z + 20z³

Taking 4z common in the equation

Now the factor is 4z(-4 + 5z²)

v) 5x²y – 15xy²

Taking 5xy common in the equation

Now the factor is 5xy(x – 3y)

vi) 10a² – 15b² + 20c²

Taking 5 common in the equation

Now the factor is 5(2a² – 3b² + 4c²)

vii) -4a2 + 4ab – 4ca

Taking 4a common in the equation

Now the factor is 4a(-a + b – c)

viii) x²yz + xy²z + xyz²

Taking xyz common in the equation

Now the factor is xyz(x + y + z)

xi) ax^{2}y + bxy^{2} + cxyz

Taking xy common in the equation

Now the factor is xy(ax + by + cz)