Factor completely $4 h^2+ g$.
A. $h (4 h+ g )$
B. $g \left(4 h^2\right)$
C. Prime
D. $\operatorname{gh}(4 h+1)$
Answer (2)
Check for common factors between 4 h 2 and g .
Verify if the expression is a difference of squares or a perfect square trinomial.
Since there are no common factors and the expression doesn't fit any standard factoring patterns, the expression is prime.
The expression 4 h 2 + g is prime and cannot be factored further: Prime .
Explanation
Understanding the Problem We are asked to factor the expression 4 h 2 + g completely. This means we want to write it as a product of simpler expressions.
Checking for Common Factors Let's examine the given expression 4 h 2 + g . We have two terms: 4 h 2 and g . We look for common factors between these terms. The term 4 h 2 has factors of 4 and h 2 (which means h × h ). The term g is just g . There are no common factors between 4 h 2 and g .
Checking for Standard Factoring Techniques Since there are only two terms and no common factors, we cannot use techniques like factoring by grouping. Also, the expression is not in the form of a difference of squares or a perfect square trinomial. Therefore, we cannot factor it using standard factoring techniques.
Conclusion Since we cannot factor the expression using any standard techniques and there are no common factors, the expression 4 h 2 + g is prime, meaning it cannot be factored further.
Examples
Factoring is a fundamental skill in algebra, used to simplify expressions and solve equations. In real life, factoring can help optimize designs, such as determining the dimensions of a garden with a specific area or calculating the most efficient way to package products. For example, if you want to build a rectangular garden with an area represented by the expression x 2 + 5 x + 6 , factoring it into ( x + 2 ) ( x + 3 ) tells you the possible dimensions of the garden.
The expression 4 h 2 + g has no common factors and does not fit any factoring patterns, so it is prime and cannot be factored further. Therefore, the correct option is C. Prime.
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