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Chapter 0: Problem 29

Factor each trinomial, or state that the trinomial is prime. $$4 x^{2}+16 x+15$$

Short Answer

Expert verified
The factorization of the trinomial \(4x^2 + 16x + 15\) is \((2x + 5)(2x + 3)\).

Step by step solution

01

Identify coefficients

First, identify the coefficients. For the trinomial \(4x^2 + 16x + 15\), 'a' is 4, 'b' is 16, and 'c' is 15.
02

Find the numbers

Now, look for two numbers that multiply to give ac (which is 4*15 = 60) and add up to give 'b' (which is 16). These two numbers are 10 and 6.
03

Rewrite the trinomial

Rewrite the trinomial by breaking up the middle term using the two numbers found: \(4x^2 + 10x + 6x + 15\).
04

Factor by grouping

Now factor by grouping. First, group the first two terms together and the last two terms together: \((4x^2 + 10x) + (6x + 15)\). Now, take out the common factor from each group: \(2x(2x + 5) + 3(2x + 5)\).
05

Factor out the common binomial

The result from step 4 is that both groups now have a binomial \(2x+5\) in common, factor it out: \((2x + 5)(2x + 3)\)

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