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Factoring quadratic expressions means breaking down a quadratic equation into simpler, multiplied factors. This often involves turning a complicated polynomial into products of binomials. For example, to factor the quadratic expression in the numerator of the first fraction in the exercise, we start with\(3d^2 + 9d - 12\). The common factor here is \(3\), so we factor that out: \(3(d^2 + 3d - 4)\). Next, we need to factor the quadratic expression \(d^2 + 3d - 4\) into two binomials. Finding numbers that multiply to -4 and sum to 3, we get: \(d^2 + 3d - 4 = (d + 4)(d - 1)\). Hence, the complete factored form is: \(3(d + 4)(d - 1)\). Knowing how to do this helps simplify the expression as it allows us to cancel out terms easily.

Once you have factored expressions, cancelling common factors becomes much easier. Cancelling involves removing identical terms from the numerator and denominator, which simplifies the fraction. In the exercise, after factoring the numerator and denominator of the first fraction, we get: \( \frac{3(d + 4)(d - 1)}{(d + 6)(d + 4)} \). Here, \( (d + 4) \) appears in both the numerator and the denominator, so we can cancel these terms out. This leaves us with: \(\frac{3(d - 1)}{d + 6}\). Cancelling common factors is essential to simplify algebraic expressions and make further multiplication or division easier.

Multiplying fractions in algebra involves multiplying the numerators together and the denominators together. Importantly, simplifying the fractions before multiplying can make the process much more straightforward. In the exercise, after simplifying the first fraction, it becomes: \( \frac{3(d - 1)}{d + 6} \). We then multiply this by the second fraction: \( \frac{d + 6}{3d} \). Multiplying the fractions involves: \( \frac{3(d - 1)(d + 6)}{(d + 6)3d} \). Before multiplying out completely, notice that the \( (d + 6) \) terms and \( 3 \) cancel out, simplifying it to: \(\frac{d - 1}{d}\). This final simplified form is what we have been aiming for. Understanding how to multiply and simplify fractions in algebra is crucial to solving more complex algebraic problems efficiently.