Factoring Trinomials
The provided trinomial is \( 4u^2 + 5u + 7 \), and the task is to factor it completely. Based on the image you provided, there are several options given, including:
1) \( u(4u + 5) + 7 \)
2) Prime
3) \( (4u - 7)(u + 5) \)
4) \( (4u + 5)(u - 7) \)
The first step in factoring is to look for a common factor. In this case, there is no common factor that divides all three terms \(4u^2\), \(5u\), and \(7\).
The next step would be to check if the trinomial can be factored into a product of two binomials of the form \( (ax + b)(cx + d) \). To do this, we would need to find two numbers that multiply to \(ac \cdot d = 4 \cdot 7 = 28\) and add up to \(b = 5\). However, there are no two integers that meet these criteria.
Therefore, this trinomial does not factor over the integers and hence is prime. The correct choice from the options provided would be:
2) Prime
