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 $a = -\frac{1}{4}$.

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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.