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Let's tackle this step by step.

### Part (a): Expanding \((x^2 + 2x - 3)(x^2 - 3x - 4)\)

We need to expand and simplify the product of these two quadratic expressions:

\[
(x^2 + 2x - 3)(x^2 - 3x - 4)
\]

To expand, we can apply the distributive property (FOIL method):

1. Multiply \(x^2\) by each term in \((x^2 - 3x - 4)\):
   \[
   x^2(x^2 - 3x - 4) = x^4 - 3x^3 - 4x^2
   \]

2. Multiply \(2x\) by each term in \((x^2 - 3x - 4)\):
   \[
   2x(x^2 - 3x - 4) = 2x^3 - 6x^2 - 8x
   \]

3. Multiply \(-3\) by each term in \((x^2 - 3x - 4)\):
   \[
   -3(x^2 - 3x - 4) = -3x^2 + 9x + 12
   \]

Now, combine all the terms:
\[
x^4 - 3x^3 - 4x^2 + 2x^3 - 6x^2 - 8x - 3x^2 + 9x + 12
\]

Group the like terms:
\[
x^4 + (-3x^3 + 2x^3) + (-4x^2 - 6x^2 - 3x^2) + (-8x + 9x) + 12
\]

Simplify:
\[
x^4 - x^3 - 13x^2 + x + 12
\]

So, we have verified that:
\[
(x^2 + 2x - 3)(x^2 - 3x - 4) = x^4 - x^3 - 13x^2 + x + 12
\]

This proves part (a).

### Part (b): Solving the equation

We are given the equation:
\[
x^4 - x^3 - 13x^2 + x + 12 = 0
\]

From part (a), we know this can be factored as:
\[
(x^2 + 2x - 3)(x^2 - 3x - 4) = 0
\]

Now, solve each quadratic equation separately.

1. \(x^2 + 2x - 3 = 0\)

Solve using the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-3)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
\]
So, \(x = 1\) or \(x = -3\).

2. \(x^2 - 3x - 4 = 0\)

Again, use the quadratic formula:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}
\]
So, \(x = 4\) or \(x = -1\).

### Final Solutions:
The solutions to the equation are:
\[
x = 1, -3, 4, -1
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you factor a cubic equation like \(x^3 + 3x^2 - 4x - 12 = 0\)?
2. What are the conditions under which a quadratic equation has real solutions?
3. How do you solve polynomial equations using synthetic division?
4. Can all quartic equations be solved by factoring? Why or why not?
5. How do you determine the nature of the roots of a quadratic equation?

### Tip:
When factoring polynomials, always check if any factorable quadratic equations can be solved easily using the quadratic formula or factoring.

Show that (x^2 + 2x - 3)(x^2 - 3x - 4) ≡ x^4 - x^3 - 13x^2 + x + 12. Hence solve the equation x^4 - x^3 - 13x^2 + x + 12 = 0.

Learn how to expand and factor the expression (x^2 + 2x - 3)(x^2 - 3x - 4) and solve the quartic equation x^4 - x^3 - 13x^2 + x + 12 = 0. This tutorial provides a step-by-step solution using the distributive property and quadratic formula. Suitable for high school students.

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