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Factor the expression completely.

10p2–5p–30

please?

1 answer

To factor the expression \(10p^2 - 5p - 30\) completely, we can follow these steps:

  1. Factor out the greatest common factor (GCF): The GCF of the coefficients (10, -5, and -30) is 5. We can factor that out first.

    \[ 10p^2 - 5p - 30 = 5(2p^2 - p - 6) \]

  2. Factor the quadratic expression \(2p^2 - p - 6\). We need to factor this quadratic into the form \((ap + b)(cp + d)\).

    To do that, we look for two numbers that multiply to \(ac = 2 \times -6 = -12\) and add up to \(b = -1\). The numbers that satisfy this condition are -4 and 3.

  3. Rewrite the middle term using -4 and 3:

    \[ 2p^2 - 4p + 3p - 6 \]

  4. Group the terms:

    \[ (2p^2 - 4p) + (3p - 6) \]

  5. Factor out the common factors from each group:

    \[ 2p(p - 2) + 3(p - 2) \]

  6. Factor out the common binomial factor \((p - 2)\):

    \[ (2p + 3)(p - 2) \]

  7. Combine with GCF:

    Now, put it all together with the GCF we factored out earlier:

    \[ 10p^2 - 5p - 30 = 5(2p + 3)(p - 2) \]

So the completely factored form of the expression \(10p^2 - 5p - 30\) is:

\[ \boxed{5(2p + 3)(p - 2)} \]