To factor the numerator \( x^2 - 5x + 50 \) of the rational expression \(\frac{x^2 - 5x + 50}{x - 10}\), Lainey should examine the quadratic \( x^2 - 5x + 50 \).
The goal of factoring is to express it in the form \((x - a)(x - b)\), where \(a\) and \(b\) are numbers that multiply to the constant term (50) and add to the linear coefficient (-5).
Now, let's analyze the choices given:
A. Divide everything by 5 - This doesn't help with factoring.
B. Keep change flip - This pertains to dividing fractions and is not relevant in factoring.
C. Look for two numbers that multiply to 50 and add to -5 - This is correct. We need to find two numbers that meet the requirements for factoring.
D. Use the differences of two squares identity - This applies to expressions like \(a^2 - b^2\) and does not apply to this quadratic expression.
Therefore, the best step Lainey should take is:
C. Look for two numbers that multiply to 50 and add to -5.
However, it's worth noting that no two integers satisfy this condition (as the quadratic has a positive discriminant and may indicate complex roots). Thus, Lainey might find that the expression cannot be factored over the integers.
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