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x-coefficient in first factor

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constant in first factor

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constant in second factor

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x-coefficient in second factor

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The first step is to find two numbers that add together to give $\\var{a*d+c*b}$ and multiply to give $\\var{a*c} \\times \\var{b*d} = \\var{a*b*c*d}$.

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These are $\\var{a*d}$ and $\\var{c*b}$.

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We split the $x$-term using $\\simplify{{a + b}x = {a} x + {b} x}$ and work as follows:

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\\\begin{aligned}\\simplify{{a*c}x^2 + {a*d + c*b} x + {b*d}} &= \\simplify{{a*c}x^2 + {a*d} x + {c*b} x + {c*d}} \\\\ &= \\simplify{{a} x({c} x + {d}) + {b}({c}x + {d})} \\\\ &= \\simplify{({a}x + {b})({c}x + {d})}\\end{aligned}\

$\\simplify{{a*c}x^2 + {a*d + c*b}x + {b*d}}$