Find each of the following products:
(i) `(x + 6) (x + 6)` (ii) `(4x + 5y) (4x + 5y)` (iii) `(7a + 9b) (7a + 9b)`
(iv) `((2)/(3) x + (4)/(5)y) ((2)/(3) x + (4)/(5) y)` (v) `(x^(2) + 7)(x^(2) + 7)` (vi) `((5)/(6) a^(2) + 2) ((5)/(6) a^(2) + 2)`

Answer

To solve the given products step by step, we will use the identity for the square of a binomial, which states that:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

Now, let's solve each part:

### (i) \((x + 6)(x + 6)\)

1. Recognize that this is the square of a binomial:
\[ (x + 6)^2 \]
2. Apply the identity:
\[ = x^2 + 2 \cdot x \cdot 6 + 6^2 \]
3. Calculate:
\[ = x^2 + 12x + 36 \]

**Final Answer for (i):** \(x^2 + 12x + 36\)

### (ii) \((4x + 5y)(4x + 5y)\)

1. Recognize that this is also the square of a binomial:
\[ (4x + 5y)^2 \]
2. Apply the identity:
\[ = (4x)^2 + 2 \cdot (4x) \cdot (5y) + (5y)^2 \]
3. Calculate:
\[ = 16x^2 + 40xy + 25y^2 \]

**Final Answer for (ii):** \(16x^2 + 40xy + 25y^2\)

### (iii) \((7a + 9b)(7a + 9b)\)

1. Recognize that this is the square of a binomial:
\[ (7a + 9b)^2 \]
2. Apply the identity:
\[ = (7a)^2 + 2 \cdot (7a) \cdot (9b) + (9b)^2 \]
3. Calculate:
\[ = 49a^2 + 126ab + 81b^2 \]

**Final Answer for (iii):** \(49a^2 + 126ab + 81b^2\)

### (iv) \(\left(\frac{2}{3}x + \frac{4}{5}y\right)\left(\frac{2}{3}x + \frac{4}{5}y\right)\)

1. Recognize that this is the square of a binomial:
\[ \left(\frac{2}{3}x + \frac{4}{5}y\right)^2 \]
2. Apply the identity:
\[ = \left(\frac{2}{3}x\right)^2 + 2 \cdot \left(\frac{2}{3}x\right) \cdot \left(\frac{4}{5}y\right) + \left(\frac{4}{5}y\right)^2 \]
3. Calculate:
\[ = \frac{4}{9}x^2 + \frac{16}{15}xy + \frac{16}{25}y^2 \]

**Final Answer for (iv):** \(\frac{4}{9}x^2 + \frac{16}{15}xy + \frac{16}{25}y^2\)

### (v) \((x^2 + 7)(x^2 + 7)\)

1. Recognize that this is the square of a binomial:
\[ (x^2 + 7)^2 \]
2. Apply the identity:
\[ = (x^2)^2 + 2 \cdot (x^2) \cdot 7 + 7^2 \]
3. Calculate:
\[ = x^4 + 14x^2 + 49 \]

**Final Answer for (v):** \(x^4 + 14x^2 + 49\)

### (vi) \(\left(\frac{5}{6}a^2 + 2\right)\left(\frac{5}{6}a^2 + 2\right)\)

1. Recognize that this is the square of a binomial:
\[ \left(\frac{5}{6}a^2 + 2\right)^2 \]
2. Apply the identity:
\[ = \left(\frac{5}{6}a^2\right)^2 + 2 \cdot \left(\frac{5}{6}a^2\right) \cdot 2 + 2^2 \]
3. Calculate:
\[ = \frac{25}{36}a^4 + \frac{20}{6}a^2 + 4 \]
Simplifying \(\frac{20}{6} = \frac{10}{3}\):
\[ = \frac{25}{36}a^4 + \frac{10}{3}a^2 + 4 \]

**Final Answer for (vi):** \(\frac{25}{36}a^4 + \frac{10}{3}a^2 + 4\)

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