Question

Find the product. $$ \frac{x^2+3 x-4}{x^2+4 x+4} \cdot \frac{2 x^2+4 x}{x^2-4 x+3} $$

Short answer

\[\frac{{2x(x+4)}}{{(x+2)(x-3)}}\]

Step-by-step solution

Factorization

Factorize all the polynomials: \[\frac{{x^2 + 3x - 4}}{{x^2 + 4x + 4}} \cdot \frac{{2x^2 + 4x}}{{x^2 - 4x + 3}} = \frac{{(x-1)(x+4)}}{{(x+2)^2}} \cdot \frac{{2x(x+2)}}{{(x-1)(x-3)}}\]

Multiplication of Numerator and Denominator

Multiply the factorized numerators with each other and the factorized denominators with each other, respectively: \[= \frac{{(x-1)(x+4) \cdot 2x(x+2)}}{{(x+2)^2 \cdot (x-1)(x-3)}}\]

Simplification

Simplify the resulting fraction by cancelling out common factors. Here, (x-1) and (x+2) cancel out from both numerator and denominator: \[= \frac{{2x(x+4)}}{{(x+2)(x-3)}}\]

Key concepts

Rational Expressions

Rational expressions are like fractions, but instead of just numbers, they contain polynomials. Think of the numerator and denominator as whole expressions made up of variables and constants. When working with rational expressions, the main goal is to simplify or solve them.
To simplify rational expressions, follow some steps:

• Factor both the numerator and the denominator.
• Look for common factors that can be cancelled out.
• Avoid cancelling out terms that aren’t factors; terms must multiply the whole numerator or denominator to cancel.

This process can make complex expressions much simpler and easier to work with. Remember that simplifying doesn’t change the value, just the form.

Factoring Polynomials

Factoring is breaking down a polynomial into a product of simpler polynomials. You might think of it like finding the building blocks of a number or expression.
Here’s how to factor polynomials effectively:

• Look for a Greatest Common Factor (GCF) in the terms and factor it out first.
• If the polynomial is quadratic, try splitting the middle term or using the quadratic formula.
• Check for special forms like difference of squares, perfect square trinomials, or sum/difference of cubes.

In our problem, each polynomial in the rational expressions was factored. For instance, \[x^2 + 3x - 4\] was turned into \((x-1)(x+4)\).
Make sure to double-check your work by expanding the factors to ensure you get the original polynomial.

Simplifying Fractions

Simplifying fractions means making them easier to understand while maintaining their value. This concept is identical with simplifying rational expressions.
Here’s how you can simplify them:

• Ensure all parts (numerator and denominator) are fully factored.
• Identify common factors between the numerator and denominator.
• Cancel out these common factors to reduce the fraction to its simplest form.

In the problem, \[(x-1)\] and \[(x+2)\] were common between the numerator and denominator.Canceling these allowed the expression to be simplified. Always check that you don’t cancel out necessary variables, as they are critical in maintaining the function’s original value.