1. (a/2a^2-7a+3) + (1/a^2+a - 12) = (a+1/2a^2+7a-4)

Let's break down and solve the given equation step by step:

The equation is:

\[
\frac{a}{2a^2 - 7a + 3} + \frac{1}{a^2 + a - 12} = \frac{a + 1}{2a^2 + 7a - 4}
\]

### Step 1: Simplify the terms.

First, we note that the equation consists of rational expressions, so we will look for a common denominator to combine the left-hand side.

- The denominators on the left side are \( 2a^2 - 7a + 3 \) and \( a^2 + a - 12 \).
- The denominator on the right side is \( 2a^2 + 7a - 4 \).

Let's attempt to factor these expressions.

#### Factor the denominators:

1. **\( 2a^2 - 7a + 3 \)**:  
   We need two numbers that multiply to \( 2 \times 3 = 6 \) and add up to \(-7\). These are \(-6\) and \(-1\).  
   So we can factor the quadratic as:
   \[
   2a^2 - 7a + 3 = (2a - 1)(a - 3)
   \]

2. **\( a^2 + a - 12 \)**:  
   We need two numbers that multiply to \(-12\) and add up to \(1\). These are \(4\) and \(-3\).  
   So we can factor the quadratic as:
   \[
   a^2 + a - 12 = (a - 3)(a + 4)
   \]

3. **\( 2a^2 + 7a - 4 \)**:  
   We need two numbers that multiply to \( 2 \times -4 = -8 \) and add up to \(7\). These are \(8\) and \(-1\).  
   So we can factor the quadratic as:
   \[
   2a^2 + 7a - 4 = (2a - 1)(a + 4)
   \]

### Step 2: Substitute the factored forms.

The equation becomes:
\[
\frac{a}{(2a - 1)(a - 3)} + \frac{1}{(a - 3)(a + 4)} = \frac{a + 1}{(2a - 1)(a + 4)}
\]

### Step 3: Combine the terms on the left side.

We need a common denominator to combine the fractions on the left. The common denominator is \( (2a - 1)(a - 3)(a + 4) \). We rewrite each term accordingly:

\[
\frac{a}{(2a - 1)(a - 3)} = \frac{a(a + 4)}{(2a - 1)(a - 3)(a + 4)}
\]
\[
\frac{1}{(a - 3)(a + 4)} = \frac{2a - 1}{(2a - 1)(a - 3)(a + 4)}
\]

Thus, the equation now becomes:
\[
\frac{a(a + 4) + (2a - 1)}{(2a - 1)(a - 3)(a + 4)} = \frac{a + 1}{(2a - 1)(a + 4)}
\]

### Step 4: Simplify the numerator on the left-hand side.

Expand \( a(a + 4) \) and combine like terms:
\[
a(a + 4) = a^2 + 4a
\]
Thus, the numerator on the left-hand side is:
\[
a^2 + 4a + 2a - 1 = a^2 + 6a - 1
\]

Now, the equation is:
\[
\frac{a^2 + 6a - 1}{(2a - 1)(a - 3)(a + 4)} = \frac{a + 1}{(2a - 1)(a + 4)}
\]

### Step 5: Eliminate the common terms.

We can cancel out the common factors \( (2a - 1) \) and \( (a + 4) \) from both sides of the equation. This simplifies to:
\[
\frac{a^2 + 6a - 1}{(a - 3)} = a + 1
\]

### Step 6: Solve the simplified equation.

Multiply both sides by \( a - 3 \) to eliminate the denominator:
\[
a^2 + 6a - 1 = (a + 1)(a - 3)
\]

Expand the right-hand side:
\[
a^2 + 6a - 1 = a^2 - 3a + a - 3
\]
\[
a^2 + 6a - 1 = a^2 - 2a - 3
\]

### Step 7: Simplify and solve for \( a \).

Cancel the \( a^2 \) terms from both sides:
\[
6a - 1 = -2a - 3
\]
Add \( 2a \) to both sides:
\[
8a - 1 = -3
\]
Add 1 to both sides:
\[
8a = -2
\]
Finally, divide by 8:
\[
a = -\frac{1}{4}
\]

### Conclusion:
The solution to the equation is \( a = -\frac{1}{4} \).

---

Would you like a more detailed breakdown of any of the steps or have any further questions about this solution?

Here are 5 follow-up questions you could explore:
1. How do you solve rational equations with multiple denominators?
2. Can you apply similar techniques to rational expressions with higher powers of \(a\)?
3. How do you check the solution to a rational equation?
4. What happens if you encounter a situation where the denominators cannot be factored?
5. How do you simplify complex rational expressions?

**Tip:** Always look for a common denominator when working with multiple fractions, as it helps combine terms efficiently and can simplify solving the equation.

Step-by-step guide to solving the rational equation involving multiple denominators. This example explains factoring quadratics and combining rational expressions. Ideal for algebra students in grades 9-11.

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