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Let's solve the provided questions step by step.

### Question 25
Solve \(2(2x + 1) = 9\)

First, distribute the 2 on the left-hand side:
\[ 4x + 2 = 9 \]

Subtract 2 from both sides:
\[ 4x = 7 \]

Divide both sides by 4:
\[ x = \frac{7}{4} \]

So, the solution is:
\[ \boxed{x = \frac{7}{4}} \]

### Question 26
Prove \((a^3+b^3) = (a+b)(a^2 - ab + b^2)\)

This is a standard algebraic identity for the sum of cubes. We can prove it by expanding the right-hand side and showing it equals the left-hand side.

Expand \((a+b)(a^2 - ab + b^2)\):
\[ (a+b)(a^2 - ab + b^2) = a(a^2 - ab + b^2) + b(a^2 - ab + b^2) \]
\[ = a^3 - a^2b + ab^2 + ba^2 - ab^2 + b^3 \]
\[ = a^3 + b^3 \]

Hence, we have proved:
\[ \boxed{a^3 + b^3 = (a+b)(a^2 - ab + b^2)} \]

### Question 27
A father is three times as old as his daughter. If the father is 36 years older than his daughter, what are their ages?

Let \( x \) be the daughter's age. Then, the father's age is \( 3x \).

Given that the father is 36 years older than the daughter:
\[ 3x = x + 36 \]

Subtract \( x \) from both sides:
\[ 2x = 36 \]

Divide both sides by 2:
\[ x = 18 \]

So, the daughter's age is 18, and the father's age is:
\[ 3 \times 18 = 54 \]

So, the solution is:
\[ \boxed{\text{Daughter's age: 18, Father's age: 54}} \]

### Question 28
Find the square root of \(x^4 + 4x^2 + 4\)

First, factorize the expression under the square root:
\[ x^4 + 4x^2 + 4 = (x^2 + 2)^2 \]

Taking the square root:
\[ \sqrt{(x^2 + 2)^2} = x^2 + 2 \]

So, the square root is:
\[ \boxed{x^2 + 2} \]

### Question 29


Explore solutions to equations, proofs of algebraic identities, and calculations of roots in this detailed math problem-solving guide. Covers equations, algebraic identities like the sum of cubes, and methods for finding roots. Suitable for high school students (Grades 9-12).

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