### Video Transcript

The area of a triangle is 12๐ฅ squared plus 38๐ฅ plus 28 centimeters squared, and its base is two ๐ฅ plus four centimeters. Find an expression for its height.

We now know that we can find the area of a triangle by multiplying the height times the base and then dividing by two, but we wanna rewrite this expression so that it says โ equals something. How do we find the height if we know the area and the base? To find that, weโll need to isolate the โ to get it by itself. Our first step would be to multiply by two on both sides.

Two divided by two is one; it cancels out. On the right side, this leaves us with height times the base equals two times the area. From there we can divide both sides of our equation by the base. ๐ divided by ๐ equals one, so two times the area divided by the base equals the height. Letโs take this formula and plug in what we know.

We know that the area equals 12๐ฅ squared plus 38๐ฅ plus 28 centimeters squared, and our base equals two ๐ฅ plus four centimeters. We notice that we have centimeters squared in the numerator and centimeters in the denominator. One of the sets of centimeters cancel out, and that reminds us that our height will be given in centimeters not in centimeters squared. We wonโt change anything about our numerator.

But for our denominator, I see a common factor of two. If I take out the factor of two, weโll reduce the expression to two times ๐ฅ plus two. And then we notice that we have a two in our numerator and in our denominator. Two divided by two equals one; these cancel each other out. But how would we divide 12๐ฅ squared plus 38๐ฅ plus 28 by ๐ฅ plus two? We need to use something called polynomial long division.

From here we ask the question what can we multiply ๐ฅ by two to equal 12๐ฅ squared. If I multiply 12๐ฅ by ๐ฅ, weโll get ๐ฅ squared. But we have to multiply our 12๐ฅ by that second term as well, so weโll have to multiply 12๐ฅ by two. From here, weโll treat the problem like any other long division problem, and weโll subtract. 12๐ฅ squared minus 12๐ฅ squared equals zero; 38๐ฅ minus 24๐ฅ equals 14๐ฅ.

We bring down our next term. Now we ask that same question again: what can we multiply ๐ฅ by to equal 14๐ฅ? Itโs 14; 14 times ๐ฅ. We canโt forget to multiply by that second term; 14 times two equals 28. We do some more subtraction here; 14๐ฅ minus 14๐ฅ equals zero; 28 minus 28 equals zero. Now we have nothing remaining. 12๐ฅ plus 14 is what we get when we divide two times the area by the base, and we can say that the height of this triangle equals 12๐ฅ plus 14 centimeters.