To factor the quadratic expression \(3x^2 + 16x + 5\), we'll look for two numbers that multiply to \(3 \cdot 5 = 15\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(16\) (the coefficient of \(x\)).
The numbers that satisfy these conditions are \(15\) and \(1\) because:
- \(15 + 1 = 16\)
- \(15 \cdot 1 = 15\)
Now, we can rewrite the middle term \(16x\) using \(15x\) and \(1x\):
\[ 3x^2 + 15x + 1x + 5 \]
Next, we group the terms:
\[ (3x^2 + 15x) + (1x + 5) \]
Now, factor out the common factors in each group:
\[ 3x(x + 5) + 1(x + 5) \]
Now, we can factor out the common binomial \((x + 5)\):
\[ (3x + 1)(x + 5) \]
Thus, the factorization of \(3x^2 + 16x + 5\) is:
\[ (3x + 1)(x + 5) \]
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