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A quadratic expression is a polynomial of degree two, which means the highest power of the variable is squared (usually represented as \( x^2 \)).
This form is generally written as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
The term "quadratic" comes from "quadratus" in Latin, meaning square.To identify a quadratic expression, take a look at the highest power of the variable.
If it's two, then you’re dealing with a quadratic expression.
In our example, \( x^2 + x - 30 \), we identify it by noting the initial \( x^2 \) term.Here, the coefficient \( a \) is 1, \( b \) is 1, and \( c \) is \(-30\).
Understanding these coefficients helps in the factoring process, particularly in deciding how to split the middle term.

Factor by grouping is a useful method for factoring certain types of polynomial expressions.
It works particularly well for expressions where grouping terms allows for the extraction of a common binomial factor.In this method:

• You first separate the expression into two groups.
• Next, factor out the greatest common factor from each group.
• After that, look for a common binomial in both parts that can be factored out.

Applying this to our example, \( x^2 + 6x - 5x - 30 \), by grouping you get \((x^2 + 6x) + (-5x - 30)\).
Factor each group separately, i.e., \( x(x + 6) \) and \(-5(x + 6)\).
Finally, you can factor out the common binomial \((x + 6)\).

A binomial factor is a polynomial with just two terms.
This factor is what’s often extracted last when you factor by grouping.Once you've separated your expression by grouping and factored each group, you should observe a common binomial factor.
For the expression \( x^2 + x - 30 \), after splitting terms and grouping effectively, we identified \((x + 6)\) as common binomial factor in the grouped terms.Extracting this common binomial leads to the completely factored form. In our example, once the \((x + 6)\) is factored out, we are left with \( (x + 6)(x - 5) \).
This step is crucial because it simplifies the quadratic expression into a product of two binomial factors, revealing its simpler "roots" or solutions.