Question

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judy wants to make a small patio in her garden. she plans to use pavers. the area of the patio is $2p^2 + 30p + 108$ square meters, and the area of each paver is $p^2 + 5p - 6$ square meters.
the number of pavers judy will need is given by the expression

Explanation:

Step1: Recall the formula for number of pavers

To find the number of pavers, we divide the area of the patio by the area of each paver. So we need to compute \(\frac{2p^{2}+30p + 108}{p^{2}+5p-6}\).

Step2: Factor the numerator and the denominator

First, factor the numerator: \(2p^{2}+30p + 108=2(p^{2}+15p + 54)=2(p + 6)(p+9)\) (since \(p^{2}+15p + 54=(p + 6)(p + 9)\) as \(6\times9 = 54\) and \(6 + 9=15\)).

Next, factor the denominator: \(p^{2}+5p-6=(p + 6)(p - 1)\) (since \(- 6=6\times(-1)\) and \(6+(-1)=5\)).

Step3: Simplify the fraction

Now, substitute the factored forms into the fraction: \(\frac{2(p + 6)(p + 9)}{(p + 6)(p - 1)}\). We can cancel out the common factor \((p + 6)\) (assuming \(p
eq - 6\) and \(p
eq1\)) to get \(\frac{2(p + 9)}{p - 1}=\frac{2p+18}{p - 1}\).

Answer:

The number of pavers is given by the expression \(\frac{2p^{2}+30p + 108}{p^{2}+5p-6}=\frac{2p + 18}{p - 1}\) (after factoring and simplifying). If we were to write the simplified form, the expression is \(\frac{2p + 18}{p - 1}\) (or \(\frac{2(p + 9)}{p - 1}\)).