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Factoring quadratics is a fundamental skill in algebra that will help you solve many types of algebraic problems. It involves breaking down a quadratic expression into simpler expressions, called factors, which when multiplied give the original expression.

Consider a quadratic in the form of \( ax^2 + bx + c \). To factor this, you must find two numbers that multiply to \( ac \) (the product of the first and last coefficients) and add to \( b \) (the middle coefficient).

For example, let's factor \( y^2 + y - 20 \):

• The product \( ac \) is \( 1 imes -20 = -20 \).
• We need two numbers that multiply to \(-20\) and add up to \(1\). These are \(5\) and \(-4\).

So the expression becomes \((y-4)(y+5)\).

By applying this method, you can factor the other quadratics: \(y^2 + 2y - 15\) as \((y-3)(y+5)\), \(y^2 + 4y - 21\) as \((y-3)(y+7)\), and \(y^2 + 3y - 28\) as \((y-4)(y+7)\). With practice, factoring will become a quick and intuitive process.

When multiplying fractions, whether they are numerical or algebraic, the process is straightforward: multiply the numerators together and then the denominators together. This rule also holds true for algebraic fractions.

Consider the expression being multiplied: \(\frac{y^{2}+y-20}{y^{2}+2y-15} \times \frac{y^{2}+4y-21}{y^{2}+3y-28}\). Once each quadratic in the numerators and denominators is factorized, the problem simplifies significantly.

After factorization, multiply the factored numerators \((y-4)(y+5)(y-3)(y+7)\) and the factored denominators \((y-3)(y+5)(y-4)(y+7)\).

Importantly, always remember to look for common factors between the numerators and denominators before proceeding to multiply, as this can drastically simplify the problem from the outset.

Simplifying algebraic expressions is a crucial step in solving equations and involves reducing expressions to their simplest form. This can often involve canceling out common factors that appear in both the numerator and the denominator of a fraction.

Once the expression is set for multiplication, after factoring the quadratics, you should look for common terms in both the numerator and the denominator. In our example, the factored form provides \((y-4)(y+5)(y-3)(y+7)\) for the numerator and \((y-3)(y+5)(y-4)(y+7)\) for the denominator.

Notice that \((y-4)\), \((y+5)\), \((y-3)\), and \((y+7)\) appear in both the numerator and denominator. Thus, these can be canceled out, resulting in the simplified expression of \(1\).

This process emphasizes the importance of factoring and recognizing common factors, which allows for simplification early on to avoid unnecessary complexity in solving the problem.