Okay, let me try to figure out how to simplify this expression: (u + 4)(u - 1) minus (u - 3)(u + 6). Hmm, where do I start? Maybe I should expand each part separately and then subtract them. Let me recall how to multiply two binomials. Oh right, the distributive property, sometimes called the FOIL method for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, right?

So first, I need to expand (u + 4)(u - 1). Let's do that step by step. The first terms are u and u, multiplying gives u². Then the outer terms: u and -1, which gives -u. The inner terms: 4 and u, which is 4u. The last terms: 4 and -1, which is -4. So when I add all those together: u² - u + 4u - 4. Wait, combining like terms: -u + 4u is 3u. So the first part simplifies to u² + 3u - 4. Okay, that seems right.

Now the second part is (u - 3)(u + 6). Let me do the same process. First: u * u is u². Outer: u * 6 is 6u. Inner: -3 * u is -3u. Last: -3 * 6 is -18. Adding those together: u² + 6u - 3u - 18. Combining like terms: 6u - 3u is 3u. So this part simplifies to u² + 3u - 18. Got that.

Now the original expression is the first part minus the second part. So that's (u² + 3u - 4) - (u² + 3u - 18). Wait, so I need to subtract each term of the second polynomial from the first. Let me write that out:

u² + 3u - 4 - u² - 3u + 18. Hmm, let's check: subtracting each term. The u² terms: u² - u² is 0. The 3u terms: 3u - 3u is also 0. The constants: -4 - (-18) is -4 + 18, which is 14. So all the u terms cancel out, leaving just 14. Wait, so the entire expression simplifies to 14? That seems surprisingly simple. Let me verify to make sure I didn't make a mistake.

Let me check the expansion of the first product again. (u + 4)(u - 1): u*u = u², u*(-1) = -u, 4*u = 4u, 4*(-1) = -4. So u² - u + 4u -4 = u² +3u -4. Correct. Then the second product: (u -3)(u +6): u*u = u², u*6 =6u, -3*u =-3u, -3*6 =-18. So u² +6u -3u -18 = u² +3u -18. Then subtracting the second product from the first: (u² +3u -4) - (u² +3u -18) = u² +3u -4 -u² -3u +18. The u² and 3u terms cancel, leaving -4 +18 =14. Yeah, that checks out. So the simplified expression is 14. Wait, but the question says "simplify", and 14 is a constant, so that's the simplest form. There's no u term left, so that's the answer. Okay, I think that's it.
Therefore, the simplified expression is $\boxed{14}$.
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