To factor the trinomial \( 5x^2 + 36x + 7 \), we can use the method of grouping or try to directly look for factors.
The trinomial is in the form \( ax^2 + bx + c \), where \( a = 5 \), \( b = 36 \), and \( c = 7 \).
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Multiply \( a \) and \( c \): \[ a \times c = 5 \times 7 = 35 \]
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We need two numbers that multiply to \( 35 \) and add to \( 36 \). The pair \( 35 \) and \( 1 \) satisfies these conditions since: \[ 35 \times 1 = 35 \quad \text{and} \quad 35 + 1 = 36 \]
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Rewrite the middle term using 35 and 1: \[ 5x^2 + 35x + 1x + 7 \]
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Group the terms: \[ (5x^2 + 35x) + (1x + 7) \]
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Factor by grouping: \[ 5x(x + 7) + 1(x + 7) \]
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Factor out the common binomial factor: \[ (5x + 1)(x + 7) \]
Thus, the factored form of \( 5x^2 + 36x + 7 \) is:
\[ \boxed{(5x + 1)(x + 7)} \]