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$\frac{-12t^2 - 18t}{8t^2 + 18}$

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Factor by grouping

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Q.

\frac{-12t^2 - 18t}{8t^2 + 18}

Factor GCF from numerator: Factor out the greatest common factor (GCF) from the numerator $-12t^2 - 18t$ . $\newline$ The GCF of $-12t^2$ and $-18t$ is $-6t$ . So we factor $-6t$ out of the numerator. $\newline$ $-12t^2 - 18t = -6t(2t + 3)$
Factor GCF from denominator: Factor out the greatest common factor (GCF) from the denominator $8t^2 + 18t$ . The GCF of $8t^2$ and $18t$ is $2t$ . So we factor $2t$ out of the denominator. $8t^2 + 18t = 2t(4t + 9)$
Rewrite with factored terms: Rewrite the expression with the factored numerator and denominator. $\newline$ The expression now looks like this: $\newline$ $(-6t(2t + 3))/(2t(4t + 9))$
Cancel common 't' term: Cancel out the common $t$ term from the numerator and denominator. $\newline$ Since $t$ is a common factor in both the numerator and the denominator, we can cancel it out, but we must be careful to not cancel it if $t = 0$ because division by zero is undefined. $\newline$ The expression simplifies to: $\newline$ $(-6(2t + 3))/(2(4t + 9))$
Simplify by canceling factors: Simplify the expression by canceling out any common numerical factors. $\newline$ We notice that $-6$ and $2$ have a common factor of $2$ . Dividing both by $2$ gives us: $\newline$ $(-3(2t + 3))/(1(4t + 9))$ $\newline$ Which simplifies to: $\newline$ $-3(2t + 3)/(4t + 9)$

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