Mind the mistake, mend the mistake

Here's a different kind of exercise. Twelve simplifications are laid out, most of them wrong. For each one: find the mistake, say what probably went wrong, and write the correct version.

−3p(−5p + 2q) is given as −3p + 5p − 2q. Whoever wrote this stopped multiplying and started adding. The −3p must multiply each term: −3p × −5p = 15p², and −3p × 2q = −6pq. Correct: 15p² − 6pq.

2(x − 1) + 3(x + 4) is given as 2x − 1 + 3x + 4. The 2 reached the x but not the −1; the 3 reached the x but not the 4. Correct: 2x − 2 + 3x + 12 = 5x + 10.

y + 2(y + 2) is turned into (y + 2)². Two quite different expressions have been muddled because they look alike (only a bracket round the whole thing would make a square). Correct: y + 2y + 4 = 3y + 4.

(5m + 6n)² is given as 25m² + 36n²: the commonest error of all. The two middle rectangles are missing. Correct: 25m² + 60mn + 36n².

5w² + 6w is combined into 11w². But w² and w aren't like terms, so nothing can be combined; 5w² + 6w is already as simple as it gets. Similarly 2a³ + 3a³ + 6a²b + 6ab² gives 5a³ correctly, but 6a²b and 6ab² are unlike and can't become 12a²b².

3a(2b × 3c) is given as 6ab × 9ac. The 3a has been multiplied in twice, once into each factor. Distributing is for a sum inside the bracket; here the bracket is a single product, 6bc, so the answer is 3a × 6bc = 18abc.

And (a + 2)(b + 4) is given as ab + 8: only the first terms and only the last terms have been multiplied. Every term must meet every term: ab + 4a + 2b + 8.

Not all twelve are wrong, though: (−q + 2)² = q² − 4q + 4 is right, and so is the last, (x + 2)(x + 5) = x² + 7x + 10, which even shows its working. Almost every slip is the same slip: something didn't reach every term.