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The equation in the image is a rational expression, where the numerator is \( 2t^2 + 7t - 4 \) and the denominator is \( -2t^2 - 5t + 3 \). 

To simplify or analyze this expression, we can attempt to factor both the numerator and the denominator, then cancel any common factors if possible.

Let me first attempt to factor both parts:

### Numerator: \( 2t^2 + 7t - 4 \)
We look for two numbers that multiply to \( 2 \times (-4) = -8 \) and add to \( 7 \). The pair of numbers that satisfy this is \( 8 \) and \( -1 \).

Thus, we rewrite the middle term:
\[
2t^2 + 8t - t - 4 = (2t^2 - t) + (8t - 4) = t(2t - 1) + 4(2t - 1) = (t + 4)(2t - 1)
\]

### Denominator: \( -2t^2 - 5t + 3 \)
We look for two numbers that multiply to \( -2 \times 3 = -6 \) and add to \( -5 \). The pair of numbers that satisfy this is \( -6 \) and \( 1 \).

Thus, we rewrite the middle term:
\[
-2t^2 - 6t + t + 3 = (-2t^2 + t) - (6t - 3) = -t(2t - 1) - 3(2t - 1) = -(t + 3)(2t - 1)
\]

### Simplified expression:
\[
\frac{(t + 4)(2t - 1)}{-(t + 3)(2t - 1)}
\]
Since \( (2t - 1) \) is common in both the numerator and the denominator, it cancels out, leaving:
\[
\frac{t + 4}{-(t + 3)} = -\frac{t + 4}{t + 3}
\]

The simplified form of the given expression is \( -\frac{t + 4}{t + 3} \).

Would you like more details on this process or have any further questions?

Here are some related questions you might want to explore:
1. How do we factor quadratic expressions in general?
2. What are the conditions for simplifying rational expressions?
3. Can this expression have undefined values?
4. How can we check if the simplification is correct?
5. What is the significance of cancelling terms in rational expressions?

**Tip:** Always check for restrictions by setting the denominator equal to zero to avoid division by zero. In this case, \( t = -3 \) would make the original expression undefined.

Simplify the rational expression given by the user: (2t^2 + 7t - 4) / (-2t^2 - 5t + 3)

This problem involves simplifying a rational expression by factoring the quadratic expressions in both the numerator and the denominator. Learn how to factor quadratics and simplify the expression step by step. Suitable for Grades 9-11.

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