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Let's go through the rational algebraic expressions one by one and simplify them.

### Problem 1:
\[
\frac{y^2 + 5x + 4}{y^2 - 3x - 4}
\]
To simplify this, we look for factorable expressions in both the numerator and the denominator.

#### Numerator: \( y^2 + 5x + 4 \)
There are no obvious factorization patterns, so we leave it as it is.

#### Denominator: \( y^2 - 3x - 4 \)
Try factoring:
- We need two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\).

So, the denominator factors as:
\[
y^2 - 3x - 4 = (y - 4)(y + 1)
\]

Thus, the simplified form is:
\[
\frac{y^2 + 5x + 4}{(y - 4)(y + 1)}
\]
This expression cannot be simplified further unless specific values for \(x\) or \(y\) are provided.

### Problem 2:
\[
\frac{-21a^2b^2}{28a^3b^3}
\]

#### Simplify the coefficients:
- \(\frac{-21}{28} = \frac{-3}{4}\)

#### Simplify the powers of \(a\) and \(b\):
- For \(a\), \(\frac{a^2}{a^3} = \frac{1}{a}\)
- For \(b\), \(\frac{b^2}{b^3} = \frac{1}{b}\)

Thus, the simplified expression is:
\[
\frac{-3}{4ab}
\]

### Problem 3:
\[
\frac{x^2 - 9}{x^2 - 7x + 12}
\]

Both the numerator and the denominator can be factored.

#### Numerator: \(x^2 - 9\) is a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

#### Denominator: \(x^2 - 7x + 12\)
- We need two numbers that multiply to \(12\) and add to \(-7\). These numbers are \(-3\) and \(-4\).
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

Thus, the expression becomes:
\[
\frac{(x - 3)(x + 3)}{(x - 3)(x - 4)}
\]

Cancel the common factor \(x - 3\) (assuming \(x \neq 3\)):
\[
\frac{x + 3}{x - 4}
\]

### Problem 4:
\[
\frac{m^2 + 6m + 5}{m^2 - m - 2}
\]

#### Factor the numerator and the denominator:

#### Numerator: \(m^2 + 6m + 5\)
- We need two numbers that multiply to \(5\) and add to \(6\). These numbers are \(1\) and \(5\).
\[
m^2 + 6m + 5 = (m + 1)(m + 5)
\]

#### Denominator: \(m^2 - m - 2\)
- We need two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\).
\[
m^2 - m - 2 = (m - 2)(m + 1)
\]

Thus, the expression becomes:
\[
\frac{(m + 1)(m + 5)}{(m - 2)(m + 1)}
\]

Cancel the common factor \(m + 1\) (assuming \(m \neq -1\)):
\[
\frac{m + 5}{m - 2}
\]

### Problem 5:
\[
\frac{x^2 - 5x - 14}{x^2 + 4x + 4}
\]

#### Factor both the numerator and the denominator:

#### Numerator: \(x^2 - 5x - 14\)
- We need two numbers that multiply to \(-14\) and add to \(-5\). These numbers are \(-7\) and \(2\).
\[
x^2 - 5x - 14 = (x - 7)(x + 2)
\]

#### Denominator: \(x^2 + 4x + 4\)
- This is a perfect square trinomial.
\[
x^2 + 4x + 4 = (x + 2)^2
\]

Thus, the expression becomes:
\[
\frac{(x - 7)(x + 2)}{(x + 2)(x + 2)}
\]

Cancel one factor of \(x + 2\) (assuming \(x \neq -2\)):
\[
\frac{x - 7}{x + 2}
\]

---

If you'd like any further details or explanations, feel free to ask!

Here are 5 related questions to deepen your understanding:

1. How do you identify when an expression can be factored?
2. What are the rules for canceling terms in rational expressions?
3. Can rational expressions always be simplified, or are there exceptions?
4. How do restrictions on variables (e.g., \(x \neq 3\)) arise in simplified forms?
5. How do you approach simplifying complex rational expressions with multiple variables?

**Tip:** Always check for common factors before canceling terms in rational expressions!

Simplify the following rational algebraic expressions: 1. (y^2 + 5x + 4) / (y^2 - 3x - 4) 2. (-21a^2b^2) / (28a^3b^3) 3. (x^2 - 9) / (x^2 - 7x + 12) 4. (m^2 + 6m + 5) / (m^2 - m - 2) 5. (x^2 - 5x - 14) / (x^2 + 4x + 4)

Simplify rational algebraic expressions involving factorization of polynomials and cancellation of common terms. Learn how to approach problems like (x^2 - 9) / (x^2 - 7x + 12) with step-by-step solutions suitable for Grades 9-11.

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