Bot GPT 4-o mini To factor the trinomial \( 5x^2 + 13x + 6 \), we will look for two numbers that multiply to \( 5 \times 6 = 30 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( 13 \) (the coefficient of \( x \)).
The numbers that work are \( 3 \) and \( 10 \), since:
- \( 3 \times 10 = 30 \)
- \( 3 + 10 = 13 \)
Now, we can rewrite the middle term (the \( 13x \)) using these two numbers:
\[ 5x^2 + 3x + 10x + 6 \]
Next, we can group the terms:
\[ (5x^2 + 3x) + (10x + 6) \]
Now, we factor by grouping:
- Factor out \( x \) from the first group and \( 2 \) from the second group:
\[ x(5x + 3) + 2(5x + 3) \]
- Now, we can factor out the common binomial \( (5x + 3) \):
\[ (5x + 3)(x + 2) \]
Thus, the factorization of the trinomial \( 5x^2 + 13x + 6 \) is:
\[ \boxed{(5x + 3)(x + 2)} \]
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