To factor the quadratic expression \(2x^2 + 5x - 3\), we are looking for two binomials of the form \((ax + b)(cx + d)\) that, when multiplied, give us the original quadratic.
The product of the coefficients of \(x^2\) (which is 2) and the constant term (which is -3) is \(2 \times -3 = -6\). We need to find two numbers that multiply to -6 and add to the coefficient of the linear term (which is 5). Those two numbers are 6 and -1, since \(6 \times -1 = -6\) and \(6 + (-1) = 5\).
Next, we rewrite the middle term using these two numbers:
\[ 2x^2 + 6x - 1x - 3 \]
Now we can factor by grouping:
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Group the first two terms and the last two terms: \[ (2x^2 + 6x) + (-1x - 3) \]
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Factor out the common factors in each group: \[ 2x(x + 3) - 1(x + 3) \]
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Now factor out the common binomial: \[ (2x - 1)(x + 3) \]
Thus, the factored form of \(2x^2 + 5x - 3\) is:
\[ (2x - 1)(x + 3) \]