Multiplying monomials by polynomials
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Multiply negative 4x squared by the whole expression 3x squared plus 25x minus 7. So if you multiply anything times a whole expression, you really just use the distributive property to multiply each term of the expression by the negative 4x squared. So we're going to have to distribute this negative 4x squared over every term in the expression. So first, we could start with negative 4x squared times 3x squared. So we can write that. We're going to have negative 4x squared times 3x squared. And to that, we're going to add negative 4x squared times 25x. And to that, we're going to add negative 4x squared times negative 7. So let's just simplify this a little bit. Now, we can obviously swap the order. We're just multiplying negative 4 times x squared times 3 times x squared. And actually, I'll do out every step. Eventually, you can do some of this in your head. This is the exact same thing as negative 4 times 3 times x squared times x squared. And what is that equal to? Well, negative 4 times 3 is negative 12. And x squared times x squared-- same base. We're taking the product. That's going to be x to the fourth. So this right here is negative 12x to the fourth. Now let's think about this term over here. This is the same thing as-- and of course, we have this plus out here. And then this part right here is the exact same thing as 25 times negative 4 times x squared times x. So let's just multiply the numbers out here. These were the coefficients. 25 times negative 4 is negative 100. So it'll plus negative 100, or we could just say it's minus 100. And then we have x squared times x, or x squared times x to the first power. Same base-- we can add the exponents. 2 plus 1 is 3. So this is negative 100x to the third power. And then let's look at this last term over here. We have negative 4x squared. So this is going to be plus-- that's this plus right over here. We have negative 4. We can multiply that times negative 7. And then multiply that times x squared. I'm just changing the order in which we multiply it. So negative 4 times negative 7 is positive 28. And then I'm going to multiply that times the x squared. There's no simplification to do, no like terms. These are different powers of x. So we are done.